tag:blogger.com,1999:blog-78731664405155696692024-03-12T21:42:16.968-07:00Jeff's BlogOriginally for a math class, and maybe some more.holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-7873166440515569669.post-35047360589893024772013-07-07T18:25:00.000-07:002013-07-07T18:25:20.031-07:00A New Statistic For Magic: The Gathering (Capstone Project)<div class="p1">
<span class="s1"><u>A New Statistic For Magic: The Gathering</u></span></div>
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<span class="s1">By: Jeff Holt</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>When we were told that we could do a project on just about anything that was related to mathematics, I immediately thought of games, and my favorite game is Magic: The Gathering. I know there are lots of math and statistics that go into MTG, but I wanted to add something to it. This is my objective: finding a new way to analyze decks and figure out an easy way to make them better. </span></div>
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<span class="s1"><u>Where to Start</u></span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>I started with some research and data gathering. I found some articles and information on counting and probabilities, which are huge when it comes to playing the game. Like any card game, people make judgements based on what they have seen, and what they have at the present time. It isn’t any different with Magic. Since Magic is a competitive game, what you’ve seen, have, and know about becomes more crucial. Your opponent can play anything within the means of their deck, and playing the probabilities becomes a mastery of a skill to have. Even the best players become burned by playing the probabilities incorrectly. Sometimes, players have to take risks to win, and many times over and over, they lose. </span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>So what can we do about it? Due to how much time I had to work on this objective, I came up with an idea that relates to economic analysis. I thought about game theory, and decision making to come up with a statistic that is, currently, a loose analyzer of how well a deck can perform based purely on color, cost, and probability. I found that an opening hand of seven cards can include certain cards, and give the probability of that combination of cards. This can be found by using the hypergeometric function, but what about the total probability of a deck? In the game, there is this idea of “curving out” a deck, or more specifically, an opening hand along with the cards you would hope to draw. Sometimes you need luck, sometimes you don’t. </span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>An ideal opening hand would include 3-4 lands, and maybe 2-3 playable spells from those lands, and maybe 1 late-game spell. Of course, as decks change, an opening hand that is perfect changes. I attempted to develop a system that can test a deck’s ability to curve out properly. There were obstacles in getting a single statistic that seemed to work, and some limited resources made getting a perfect outcome impossible. Despite this, I came up with a statistic that I call the “Curve Efficiency Rating”. This statistic calculates the total probability of cards that can be drawn, along with being able to play them as soon as possible. </span></div>
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<span class="s1"><u>Inspiration and Details</u></span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>One of the articles I read discussed the usefulness of playing fetch lands in Magic. Fetch lands are a nickname for a set of lands that are played like a normal land, but are sacrificed and allows you to search for another land that is of the colors that you need to cast spells in your hand. </span></div>
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<a href="http://4.bp.blogspot.com/-onjR_iu8fVo/UdoR16gxN2I/AAAAAAAAAEY/D-FnhCHSFjY/s1600/graph2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="427" src="http://4.bp.blogspot.com/-onjR_iu8fVo/UdoR16gxN2I/AAAAAAAAAEY/D-FnhCHSFjY/s640/graph2.jpg" width="640" /></a></div>
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<span class="s1"><span class="Apple-tab-span"> </span>This was part of a study by Garrett Johnson that found the effectiveness of using fetch lands in a deck. The probabilities of drawing lands in your opening hand change by how many lands you expect to start with. (The legend in the graph represents the number of fetches in a deck). </span></div>
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<a href="http://1.bp.blogspot.com/-XBJtfmbsU4o/UdoR1siu7II/AAAAAAAAAEc/rOTNJc9K5Cw/s1600/graph1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="438" src="http://1.bp.blogspot.com/-XBJtfmbsU4o/UdoR1siu7II/AAAAAAAAAEc/rOTNJc9K5Cw/s640/graph1.jpg" width="640" /></a></div>
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<span class="s1"><span class="Apple-tab-span"> </span>On average, a 60 card deck with 24 lands will start with 2.8 lands in each opening hand. This kind of simple statistics was what I wanted to try to find. </span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>The Curve Efficiency Rating is composed of a summation of probabilities of cards in a deck. To start, I looked at 20 of the most popular decks in the Standard format, (this format allows players to construct decks using cards from the last 2 blocks that have been released, with no more than 4 copies of a single card in a deck, besides basic lands). When gathering data, I wanted to focus on land types, converted mana costs (which is the total number of mana needed to cast a spell), and the colors of the spells. This is a basic summary of the decks I looked at:</span></div>
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<br /></div>
<table cellpadding="0" cellspacing="0" class="t1">
<tbody>
<tr>
<td class="td1" valign="top">
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<span class="s1"><b>Deck Names</b></span></div>
</td>
<td class="td1" valign="top">
<div class="p3">
<span class="s1"><b> Total Lands</b></span></div>
</td>
<td class="td1" valign="top">
<div class="p3">
<span class="s1"><b> Total Creatures</b></span></div>
</td>
<td class="td1" valign="top">
<div class="p3">
<span class="s1"><b> Total Non-Creatures</b></span></div>
</td>
<td class="td2" valign="top">
<div class="p3">
<span class="s1"><b> Total Multi-Colored Spells</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p2">
<br /></div>
</td>
<td class="td3" valign="top">
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<br /></div>
</td>
<td class="td3" valign="top">
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<br /></div>
</td>
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<br /></div>
</td>
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<br /></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Jund </b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 25</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>6</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>17</b></span></div>
</td>
<td class="td4" valign="top">
<div class="p4">
<span class="s1"><b>12</b></span></div>
</td>
</tr>
<tr>
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<div class="p3">
<span class="s1"><b>Reanimator</b></span></div>
</td>
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<span class="s1"><b> 23</b></span></div>
</td>
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<div class="p4">
<span class="s1"><b>22</b></span></div>
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<div class="p4">
<span class="s1"><b>9</b></span></div>
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<div class="p4">
<span class="s1"><b>6</b></span></div>
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</tr>
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<div class="p3">
<span class="s1"><b>Junk Crats</b></span></div>
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<span class="s1"><b> 25</b></span></div>
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<div class="p4">
<span class="s1"><b>13</b></span></div>
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<div class="p4">
<span class="s1"><b>11</b></span></div>
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<div class="p4">
<span class="s1"><b>11</b></span></div>
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<div class="p3">
<span class="s1"><b>Bant Hex</b></span></div>
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<span class="s1"><b> 22</b></span></div>
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<div class="p4">
<span class="s1"><b>11</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>13</b></span></div>
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<div class="p4">
<span class="s1"><b>14</b></span></div>
</td>
</tr>
<tr>
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<div class="p3">
<span class="s1"><b>UWR Mid</b></span></div>
</td>
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<div class="p4">
<span class="s1"><b> 25</b></span></div>
</td>
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<div class="p4">
<span class="s1"><b>15</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>13</b></span></div>
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<div class="p4">
<span class="s1"><b>7</b></span></div>
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<span class="s1"><b>Naya</b></span></div>
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<span class="s1"><b> 24</b></span></div>
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<div class="p4">
<span class="s1"><b>12</b></span></div>
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<div class="p4">
<span class="s1"><b>3</b></span></div>
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<div class="p4">
<span class="s1"><b>21</b></span></div>
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<div class="p3">
<span class="s1"><b>R/G Aggro</b></span></div>
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<span class="s1"><b> 23</b></span></div>
</td>
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<div class="p4">
<span class="s1"><b>16</b></span></div>
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<div class="p4">
<span class="s1"><b>10</b></span></div>
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<div class="p4">
<span class="s1"><b>12</b></span></div>
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<div class="p3">
<span class="s1"><b>UWR Flash</b></span></div>
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<span class="s1"><b> 26</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>10</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>10</b></span></div>
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<div class="p4">
<span class="s1"><b>14</b></span></div>
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<span class="s1"><b>Esper Control</b></span></div>
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<span class="s1"><b> 27</b></span></div>
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<div class="p4">
<span class="s1"><b>5</b></span></div>
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<div class="p4">
<span class="s1"><b>16</b></span></div>
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<div class="p4">
<span class="s1"><b>12</b></span></div>
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<span class="s1"><b>B/G Midrange</b></span></div>
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<span class="s1"><b> 24</b></span></div>
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<div class="p4">
<span class="s1"><b>14</b></span></div>
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<div class="p4">
<span class="s1"><b>18</b></span></div>
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<div class="p4">
<span class="s1"><b>4</b></span></div>
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<span class="s1"><b>Jund Aggro</b></span></div>
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<span class="s1"><b> 23</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>12</b></span></div>
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<div class="p4">
<span class="s1"><b>3</b></span></div>
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<div class="p4">
<span class="s1"><b>26</b></span></div>
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<div class="p3">
<span class="s1"><b>Prime Bant</b></span></div>
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<span class="s1"><b> 24</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>14</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>9</b></span></div>
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<div class="p4">
<span class="s1"><b>13</b></span></div>
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<div class="p3">
<span class="s1"><b>Naya Blitz</b></span></div>
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<div class="p4">
<span class="s1"><b> 21</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>32</b></span></div>
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<span class="s1"><b>0</b></span></div>
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<span class="s1"><b>7</b></span></div>
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<span class="s1"><b>B/G Zombies </b></span></div>
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<span class="s1"><b> 22</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>12</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>10</b></span></div>
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<div class="p4">
<span class="s1"><b>16</b></span></div>
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<div class="p3">
<span class="s1"><b>Act 2</b></span></div>
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<div class="p4">
<span class="s1"><b> 24</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>8</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>11</b></span></div>
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<div class="p4">
<span class="s1"><b>17</b></span></div>
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<div class="p3">
<span class="s1"><b>UWR Geist</b></span></div>
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<div class="p4">
<span class="s1"><b> 24</b></span></div>
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<div class="p4">
<span class="s1"><b>9</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>10</b></span></div>
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<div class="p4">
<span class="s1"><b>17</b></span></div>
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<div class="p3">
<span class="s1"><b>Bant Flash</b></span></div>
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<div class="p4">
<span class="s1"><b> 25</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>6</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>11</b></span></div>
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<td class="td4" valign="top">
<div class="p4">
<span class="s1"><b>18</b></span></div>
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<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Naya Humans</b></span></div>
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<div class="p4">
<span class="s1"><b> 20</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>32</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>2</b></span></div>
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<td class="td4" valign="top">
<div class="p4">
<span class="s1"><b>6</b></span></div>
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</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>4-C Rites</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 23</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>8</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>19</b></span></div>
</td>
<td class="td4" valign="top">
<div class="p4">
<span class="s1"><b>10</b></span></div>
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<div class="p3">
<span class="s1"><b>BWR Mid</b></span></div>
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<div class="p4">
<span class="s1"><b> 25</b></span></div>
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<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>3</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b>9</b></span></div>
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<div class="p4">
<span class="s1"><b>23</b></span></div>
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<span class="s1"><b><br /></b></span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>Each deck has 60 cards, except for R/G Aggro (which has 61), and Jund Aggro (64). The actual dataset contains all cards broken down into these categories, but then broken into their colors and mana costs. What the Curve Efficiency Rating calculates is the total summation of the probability of drawing a particular spell, and the lands needed to cast it. </span></div>
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<span class="s1"><span class="Apple-tab-span"> </span>Each card has a probability of being drawn, that is calculated with the chances of drawing the land needed to play it on the earliest turn. There are a few things that are taken into consideration when making these calculations:</span></div>
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<span class="s1"></span><br /></div>
<ol class="ol1">
<li class="li1"><span class="s1">Start with a 7 card hand, and be on the play. This means that you start with 7 cards, and do not draw until the second turn. </span></li>
<li class="li1"><span class="s1">There are no other effects that take place. Mana ramp is not calculated, neither are the plays of an opponent. It is just the raw casting cost and being able to cast it. </span></li>
<li class="li1"><span class="s1">The first turn means turn one, so we want to play a one-cost card on turn one.</span></li>
<li class="li1"><span class="s1">All spells with an X in the casting cost have an X of 3. So a card like Sphinx’s Revelation, that costs WUUX, (one white mana, two blue mana, and X additional mana) will cost 6 mana total in this study. </span></li>
<li class="li1"><span class="s1">Multi-colored spells are generalized and grouped together.</span></li>
<li class="li1"><span class="s1">Each spell is considered its own “game”, such that we consider each spell individually, and it’s purely its own chance of being played without being effected by anything else. </span></li>
</ol>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>Let’s look into the first deck: Jund. Jund plays black, green, and red spells. It has 25 lands: 8 green/black, 7 black/red, and 7 green/red dual lands, and 3 other lands that only produce 1 colorless mana each turn. Consider a spell that costs 1 black mana to play, what are the chances that we have that spell and the mana to cast it on turn? There are 15 black-mana sources, so we need one of the black sources, and the spell itself. If there are 4 copies of the spell, then the hypergeometric equation will give us a probability of having the cards we need to play that spell on turn one. </span></div>
<div class="p1">
<span class="s1"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-ZAmu_RYibKc/UdoR1z5nNTI/AAAAAAAAAEU/JsFz2Ji2Gbc/s1600/num.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="142" src="http://3.bp.blogspot.com/-ZAmu_RYibKc/UdoR1z5nNTI/AAAAAAAAAEU/JsFz2Ji2Gbc/s200/num.jpg" width="200" /></a></div>
<div class="p1">
<span class="s1"><br /></span></div>
<div class="p1">
<span class="s1"><br /></span></div>
<div class="p1">
<span class="s1">Where: </span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>N = the deck size</span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>m = the card we want (number of copies)</span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>n = the remaining number of cards drawn</span></div>
<div class="p1">
<span class="s1">
</span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>k = the card we need from the number of copies (always 1)</span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"><span class="Apple-tab-span"> </span></span></div>
<div class="p2">
<span class="s1"><span class="Apple-tab-span"> </span></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>We take the basic example of the equation, and multiply it by each factor that goes into what we want. We need 1 of the 4 copies of a card to play it, plus 1 black of the 15 to cast it on turn one. We also will have 5 more cards, but they can be anything that is not needed, so the equation will look something like this for this spell, (sorry for the quality, my original image wouldn't work):</span></div>
<div class="p2">
<br /><span class="s1"></span></div>
<div class="p2">
(15) (4) (41)</div>
<div class="p2">
<span class="s1">( 1) (1) (5)</span></div>
<div class="p2">
<u> </u></div>
<div class="p2">
(60)</div>
<div class="p2">
(7)</div>
<div class="p2">
<br /></div>
<div class="p2">
</div>
<div class="p1">
<span class="s1">and so the probability of being able to play this spell on turn one is equal to 11.64%. I’ve calculated the probability for each card in each deck by using this same formula. The only difference from card to card is multiplying by more mana probabilities as they are needed (such as sheer number of mana to different colors). Then I add each probability together for each card in a deck, and get the Curve Efficiency Rating. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><u>Analyzing the CER</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>It’s quite simple; the higher the number, the more efficient the deck is. Decks that are controlling (meaning blue decks usually), have a higher rating because their spells are more expensive. This means they have more time to draw them, and the lands needed, to play them on the earliest turn. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<br /></div>
<table cellpadding="0" cellspacing="0" class="t1">
<tbody>
<tr>
<td class="td5" valign="top">
<div class="p3">
<span class="s1"><b>Deck Names</b></span></div>
</td>
<td class="td5" valign="top">
<div class="p4">
<span class="s1"><b> curve efficiency rating</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p2">
<br /></div>
</td>
<td class="td3" valign="top">
<div class="p2">
<br /></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Jund</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 2.2839</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Reanimator</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.37678</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Junk Crats</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.13647</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Bant Hex</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.03669</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>UWR Mid</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 2.29767</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Naya</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.48565</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>R/G Aggro</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.69614</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>UWR Flash</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 3.02176</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Esper Control</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 3.08138</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>B/G Midrange </b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 2.19277</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Jund Aggro</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.25698</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Prime Bant</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.87719</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Naya Blitz</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 0.6847</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>B/G Zombies </b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 0.93898</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Act 2</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.64747</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>UWR Geist</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 4.16444</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Bant Flash</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 2.13506</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>Naya Humans</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.75087</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>4-C Rites</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 1.25076</b></span></div>
</td>
</tr>
<tr>
<td class="td3" valign="top">
<div class="p3">
<span class="s1"><b>BWR Mid</b></span></div>
</td>
<td class="td3" valign="top">
<div class="p4">
<span class="s1"><b> 2.70770</b></span></div>
</td>
</tr>
</tbody>
</table>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>If we understand deck compositions, we see that aggressive decks have low ratings, and this is because many of their spells are low-costing. The midrange decks have spells that are mostly in the 3-5 mana cost range, and the blue decks (of course) have the best ratings because they have higher costing cards. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><u>What This Leads To</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>Economic analysis uses regressional outputs to predict financial means for demographics amongst the population. Using a similar process, I have built 4 regression models to help deck-builders analyze their composition. Using a small-variant regression output in MS Excel, I found these models based on land, creature spells, non-creature spells, and multicolored spells: </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<br /></div>
<table cellpadding="0" cellspacing="0" class="t1">
<tbody>
<tr>
<td class="td6" valign="bottom">
<div class="p5">
<span class="s1"><i>Land </i></span></div>
</td>
<td class="td7" valign="bottom">
<div class="p5">
<span class="s1"><i>Coefficients</i></span></div>
</td>
</tr>
<tr>
<td class="td8" valign="bottom">
<div class="p6">
<span class="s1">Intercept </span></div>
</td>
<td class="td9" valign="bottom">
<div class="p7">
<span class="s1">-0.72739</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">U/W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.0299</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">W/G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.186317</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">G/B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.143789</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">B/R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.309508</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">R/U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.293637</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">R/W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.22147</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">W/B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.05554</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">B/U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.565639</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">U/G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.13486</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">G/R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.06993</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.96358</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.365084</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.090531</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.02711</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.201999</span></div>
</td>
</tr>
<tr>
<td class="td12" valign="bottom">
<div class="p6">
<span class="s1">Land</span></div>
</td>
<td class="td13" valign="bottom">
<div class="p7">
<span class="s1">-0.02208</span></div>
</td>
</tr>
</tbody>
</table>
<br /><div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>This is the model for lands. They include dual lands, basics, and non-basic/dual lands that only produce colorless mana. Each coefficient is multiplied by the respective number of lands of that type, and then added together with the intercept value. Lands that are positive add more the CER, while negative values do not. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<br /></div>
<table cellpadding="0" cellspacing="0" class="t1">
<tbody>
<tr>
<td class="td6" valign="bottom">
<div class="p5">
<span class="s1"><i>Creature</i></span></div>
</td>
<td class="td7" valign="bottom">
<div class="p5">
<span class="s1"><i>Coefficients</i></span></div>
</td>
</tr>
<tr>
<td class="td8" valign="bottom">
<div class="p6">
<span class="s1">Intercept </span></div>
</td>
<td class="td9" valign="bottom">
<div class="p7">
<span class="s1">2.837139</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.06215</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.26636</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.17733</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.08536</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.28486</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.18203</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.18532</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.11453</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.04863</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.036418</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.058173</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.82755</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">4-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.349107</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">4-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.04141</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">4-R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0</span></div>
</td>
</tr>
<tr>
<td class="td12" valign="bottom">
<div class="p6">
<span class="s1">5-G</span></div>
</td>
<td class="td13" valign="bottom">
<div class="p7">
<span class="s1">-0.28138</span></div>
</td>
</tr>
</tbody>
</table>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"><span class="Apple-tab-span"> </span></span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>This is the creature model. When a variable says “1-W”, that means it is a white spell that requires one mana to cast (thus, that one mana must be white). A “2-U” creature is a blue spell that requires two mana (one being blue), and so. The same idea is made in regards to the coefficients for each variable as before. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<br /></div>
<table cellpadding="0" cellspacing="0" class="t1">
<tbody>
<tr>
<td class="td14" valign="bottom">
<div class="p5">
<span class="s1"><i>Non-creature</i></span></div>
</td>
<td class="td15" valign="bottom">
<div class="p5">
<span class="s1"><i> Coefficients</i></span></div>
</td>
</tr>
<tr>
<td class="td8" valign="bottom">
<div class="p6">
<span class="s1">Intercept </span></div>
</td>
<td class="td9" valign="bottom">
<div class="p7">
<span class="s1">1.419468</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.20993</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.446506</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.191473</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">1-R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.445932</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-2.65702</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.33504</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.37775</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-G</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.08625</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-R</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.10049</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">2-C</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-W</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.45677</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-U</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.575053</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-B</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.269095</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">3-C</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.34872</span></div>
</td>
</tr>
<tr>
<td class="td12" valign="bottom">
<div class="p6">
<span class="s1">4-U</span></div>
</td>
<td class="td13" valign="bottom">
<div class="p7">
<span class="s1">0.493799</span></div>
</td>
</tr>
</tbody>
</table>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"><span class="Apple-tab-span"> </span></span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>This is the model for non-creature spells, and it follows the same logic as creature spells. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<br /></div>
<table cellpadding="0" cellspacing="0" class="t1">
<tbody>
<tr>
<td class="td16" valign="bottom">
<div class="p5">
<span class="s1"><i>Multi-colored spells</i></span></div>
</td>
<td class="td17" valign="bottom">
<div class="p5">
<span class="s1"><i>Coefficients</i></span></div>
</td>
</tr>
<tr>
<td class="td8" valign="bottom">
<div class="p6">
<span class="s1">Intercept </span></div>
</td>
<td class="td9" valign="bottom">
<div class="p7">
<span class="s1">1.937248</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">MC-1</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.05796</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">MC-2</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">-0.12051</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">MC-3</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.00901</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">MC-4</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.040001</span></div>
</td>
</tr>
<tr>
<td class="td10" valign="bottom">
<div class="p6">
<span class="s1">MC-5</span></div>
</td>
<td class="td11" valign="bottom">
<div class="p7">
<span class="s1">0.069495</span></div>
</td>
</tr>
<tr>
<td class="td12" valign="bottom">
<div class="p6">
<span class="s1">MC-6</span></div>
</td>
<td class="td13" valign="bottom">
<div class="p7">
<span class="s1">0.335003</span></div>
</td>
</tr>
</tbody>
</table>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"><span class="Apple-tab-span"> </span></span></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>Finally, this is the multi-colored spell model. It follows the same logic as the previous models. A deck-builder can use these models to figure out their decks efficiency on casting the cards in the deck. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><u>Problems</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>For one, it can’t predict games. That sort of variance is impossible to calculate on a game-to-game basis. There is too much that goes on in each game, and that’s what makes it so fun to play. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>Secondly, the regression model needs tuning. I did not have access to the best tools to run a 50+ variable regression, and had to break it up into smaller models. While the numbers may not line up to the total rating, it’s the same idea that the higher the total number between the four models, the better the deck will be at casting cards as soon as possible. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>Thirdly, calculating individual multi-colored spells would provide a more accurate model. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><u>Where Could It Go From Here</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>I’d love to work more with this kind of math involved with the game. Everyone knows of saber-metrics for baseball, and the increase in better-analyzing statistics for the sport of basketball. Working with these kinds of numbers would be so much fun for me, and I’d get to be involved with a game that is big part of my personal (and now somewhat academic) life. If I put the time into this sort of work more, then I’m sure it’s possible to break down all sorts of ratings for cards, rather than using hours of play-testing to figure out if one spell should be in a deck over another. These kinds of statistics can further prove that a card is better over another, although the natural fun of picking cards and playing decks will never be replaced by number-crunching. I’d love to expand this same kind of work to other formats, and have an eternal database for legacy, modern, and more popular EDH decks (since those are formats that do not change nearly as often as standard does). As for a project now, I couldn’t have thought of a more enjoyable topic to research and study. Getting to watch games, read articles, and study the calculations behind this game is complete bliss. I suppose a distant dream would be to do this sort of work as a part-time career, but I’ve enjoyed the entire process of what I’ve done so far.</span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">Thank you for reading. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">Sources:</span></div>
<div class="p8">
<span class="s2"><a href="http://magic.tcgplayer.com/db/print.asp?ID=3096">http://magic.tcgplayer.com/db/print.asp?ID=3096</a></span></div>
<div class="p8">
<span class="s2"><a href="http://www.gatheringmagic.com/chrismascioli-100512-of-math-and-magic-part-1-the-hypergeometric-distribution/">http://www.gatheringmagic.com/chrismascioli-100512-of-math-and-magic-part-1-the-hypergeometric-distribution/</a></span></div>
<br />
<div class="p8">
<span class="s2"><a href="http://www.kibble.net/magic/magic09.php">http://www.kibble.net/magic/magic09.php</a></span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com3tag:blogger.com,1999:blog-7873166440515569669.post-86288731373440011272013-07-07T13:10:00.000-07:002013-07-07T13:18:36.248-07:00Weekly Exemplars Week 6 (History)<br />
<br />
I chose this as my history selection since it is a comprehensive description of the game, and the mathematician (Richard Garfield) that invented the game. The game is very fun for me, and was the easiest subject matter to go into detail on.<br />
<br />
Week 7 (Doing)<br />
<br />
I chose this as my doing selection because of the fun I had in researching paradoxes, and the Rubik's Cube, as well as expanding topics from a fun book. The Rubik's Cube was always challenging, but now it's easier to understand<br />
<br />
Week 3 (Nature)<br />
<br />
I chose this as my nature selection because algebra is a vast subject, and it was simple to dissect the differences between it and other branches of mathematics.<br />
<br />
Week 8 (Communication)<br />
<br />
I chose this as my communication selection because the book I read was awesome. I was able to to do a joint weekly article that was about The Math Book, and discussing it in a review form was enjoyable.<br />
<br />
<br />
To view these, go to my blog page, and you will find all the articles. <a href="http://www.blogger.com/blogger.g?blogID=7873166440515569669#allposts/postNum=4">http://www.blogger.com/blogger.g?blogID=7873166440515569669#allposts/postNum=4</a><br />
<div class="p4">
</div>
<br />
<br />
<br />holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-6145475936995029922013-07-07T12:55:00.001-07:002013-07-07T12:55:32.380-07:00Book 1 Notes - The Book of Modern Math<div class="p1">
<span class="s1"><u>Chapter 3</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">Counting numbers -> whole numbers -> integers -> rational -> real -> complex</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span> ->fractions /</span></div>
<div class="p1">
<span class="s2">systems of numbers can be similar to what we know as modulo systems</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span>in the book, consider assigning numbers to the days of week, where </span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Sunday = 0</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Monday = 1</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Tuesday = 2</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Wednesday = 3</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Thursday = 4</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Friday = 5</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Saturday = 6</span></div>
<div class="p1">
<span class="s2">and then add/subtract 7 from that number, and you can form equations to identify days in a week. </span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Sunday = 7n</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>Monday = 7n+1...and so</span></div>
<div class="p1">
<span class="s2">this allows for higher counting in arithmetic, the development of addition and multiplication tables, and these are further known as congruence classes.</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">theorem: x^(p-1) = 1 (mod p) for any prime p and any x not congruent to 0 (mod p)</span></div>
<div class="p1">
<span class="s2">Wilson’s theorem: 1, 2, 3,...(p-1) = -1 (mod p)</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 4</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">sets have developed their own language that represents what is and isn’t part of a set, the book emphasizes the use of {} to represent being in a set. If a number or object is in the brackets, then it is part of a set. Otherwise, it would not be part of the set.</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">the empty is represented by the cross of a “/” and “0/”</span></div>
<div class="p1">
<span class="s2">if two sets have the same content, then they are equal</span></div>
<div class="p1">
<span class="s2">subsets are parts of a whole set</span></div>
<div class="p1">
<span class="s2">intersections are objects that belong to two sets, and usually there are objects that belong in the individual sets. This is represented by a venn diagram</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 5</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">functions and formulas usually involve x as an independent variable, where x cannot have multiple outputs, but it is possible for multiple inputs to have the same output.</span></div>
<div class="p1">
<span class="s2">an interesting concept is a function within another functions, because numbers in the first function, must have outputs that are in the domain of the second function</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>g(f(x))</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">a function is defined on some set</span></div>
<div class="p1">
<span class="s2">it takes values in some set</span></div>
<div class="p1">
<span class="s2">it is defined if a rule is known for finding a value for a given input, and if it’s unique</span></div>
<div class="p1">
<span class="s2">bijections, or one-to-one correspondence, are exactly those functions which have intervene functions</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 6</u></span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">abstract algebra involves many axioms, which are statements that are known to be true, but not proven to be true</span></div>
<div class="p1">
<span class="s2">rings, fields, and domains use axioms for all kinds of proofs</span></div>
<div class="p1">
<span class="s2">an interesting aspect in the set of unions, where intersections add or subtract a sum of combinations</span></div>
<div class="p1">
<span class="s2"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-AeAd059CzbQ/UdnHpoIe5kI/AAAAAAAAAD8/4WG_9TiWiOg/s1600/venn.001.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-AeAd059CzbQ/UdnHpoIe5kI/AAAAAAAAAD8/4WG_9TiWiOg/s320/venn.001.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="p1">
<span class="s2"><br /></span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">the sum of combinations would be: A + B + C - D - E - F + G</span></div>
<div class="p1">
<span class="s2">you add the individual section, subtract intersections of two sets, add intersections of three sets, and it continues alternating to account for the intersection of other sets</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">consider mod 7</span></div>
<div class="p1">
<span class="s2">[5] + [4] = [2]</span></div>
<div class="p1">
<span class="s2">[6] x [3] = [4]</span></div>
<div class="p1">
<span class="s2">congruence is fun</span></div>
<div class="p1">
<span class="s2">complex numbers involve i, where i^2 = -1</span></div>
<div class="p1">
<span class="s2"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-v1C1BGjvXsk/UdnGF6rQgJI/AAAAAAAAADo/YOUf4bgpIec/s1600/table.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="147" src="http://3.bp.blogspot.com/-v1C1BGjvXsk/UdnGF6rQgJI/AAAAAAAAADo/YOUf4bgpIec/s320/table.png" width="320" /></a></div>
<div class="p1">
<span class="s2"><br /></span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">this are the addition and multiplication tables for the congruence class of 4</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 7</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">symmetry is the basic idea that dissecting an object or image or anything and that both sides are the same in some way, whether they are mirror images, or show the same pattern after a rotation, etc.</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 8 </u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">Euclid’s Axioms:</span></div>
<ol class="ol1">
<li class="li1"><span class="s2">any two points lie on a straight line</span></li>
<li class="li1"><span class="s2">two lines meet in at most two points</span></li>
<li class="li1"><span class="s2">any finite line segment may be produced as far as you wish</span></li>
<li class="li1"><span class="s2">it is possible to describe a circle with any center and any radius</span></li>
<li class="li1"><span class="s2">all right angles are equal</span></li>
<li class="li1"><span class="s2">given any line, and any point not on the line, then there exists exactly one line parallel to the first line and passing through the given point</span></li>
</ol>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">Hilbert gave two requirements for an axiomatic system; completeness, and independence</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 9</u></span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">cardinal traits</span></div>
<ol class="ol1">
<li class="li1"><span class="s2">a<=a for any cardinal a</span></li>
<li class="li1"><span class="s2">if a<=B, and B<=y, then a<=y</span></li>
<li class="li1"><span class="s2">if a<=B, and B<=a, then a=B</span></li>
</ol>
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<span class="s2"></span><br /></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 10</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">“Topology is the study of those properties of geometrical objects which remain unchanged under continuous transformations of the object”.</span></div>
<div class="p1">
<span class="s2">basic objects in topology are topological spaces</span></div>
<div class="p1">
<span class="s2">if two topological spaces can be passed from one to the other in a continuous way, and back in a continuous way, then it is considered topologically equivalent </span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 11</u></span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">Graph theory is the idea behind networks and intersections.</span></div>
<div class="p1">
<span class="s2">A K_3,3 graph cannot be written/designed without any of the edges not intersecting</span></div>
<div class="p1">
<span class="s2">A network has two main parts: 1. a set, N, whole elements are called nodes or vertices, 2. a way of specifying when two vertices are joined together</span></div>
<div class="p1">
<span class="s2">Euler’s Formula: V - E + F = 2</span></div>
<div class="p1">
<span class="s2">The four color theorem cannot be proven sufficiently, but the five color theorem has been proven to be enough for sections in graph theory. It’s weird since it always seems that four colors is enough to have it so no two areas border each other with the same color</span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 12</u></span></div>
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<span class="s1"></span><br /></div>
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<span class="s2">Topological properties: closed curve, connected, disconnected; these prove spheres and toruses are topologically distinct</span></div>
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<span class="s2"><br /></span></div>
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<a href="http://3.bp.blogspot.com/-7wkk_4dwlfA/UdnGF-M4qOI/AAAAAAAAADk/TUqwUNda4-U/s1600/torus.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="221" src="http://3.bp.blogspot.com/-7wkk_4dwlfA/UdnGF-M4qOI/AAAAAAAAADk/TUqwUNda4-U/s320/torus.png" width="320" /></a></div>
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<span class="s2"><br /></span></div>
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<span class="s2"></span><br /></div>
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<span class="s2">torus</span></div>
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<span class="s2">The sphere, torus, and double torus are orientable</span></div>
<div class="p1">
<span class="s2">The projective plane, and Klein bottle are not orientable</span></div>
<div class="p1">
<span class="s2">Topology has a process called “surgery” where spaces were cut up and put back together again</span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 13</u></span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">Holes are gaps in topological spaces</span></div>
<div class="p1">
<span class="s2">Paths are lines that connect two points of a topological space across the space</span></div>
<div class="p1">
<span class="s2">paths can be used to define functions</span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 14</u></span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">1-space (first dimension) = R = the set of real numbers, x</span></div>
<div class="p1">
<span class="s2">2-space (second) = R^2 = the set of pairs of real numbers (x,y)</span></div>
<div class="p1">
<span class="s2">3-space (third) = R^3 = the set of triples of real numbers (x,y,z)</span></div>
<div class="p1">
<span class="s2">since the 4th dimension is not defined, there isn’t any labeling for it yet</span></div>
<div class="p1">
<span class="s2">the issue comes with “distance”, which is found in the first three dimensions but not in the fourth</span></div>
<div class="p1">
<span class="s2">distance has three conditions</span></div>
<ol class="ol1">
<li class="li1"><span class="s2">the distance between two points is positive</span></li>
<li class="li1"><span class="s2">the distance between two points is the same in either direction</span></li>
<li class="li1"><span class="s2">the distance from A to B should not be linger than A to C plus the distance from C to B</span></li>
</ol>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">the 5 regular polyhedra in the 3-space are: tetrahedron, cube, octahedron, dodecahedron, icosahedron.</span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 15</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">Linear algebraically systems have either one solution, no solutions, or infinitely many solutions</span></div>
<div class="p1">
<span class="s2">matrices are of the form:</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
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<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(a b)</span></div>
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<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(c d)</span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">and vectors look like:</span></div>
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<span class="s2"></span><br /></div>
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<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(x)</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(y)</span></div>
<div class="p1">
<span class="s2">and multiplying a matrix by a vector can become another vector or matrix</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(a b) (x) = (X) = (ax + by)</span></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(c d) (y) (Y) (cx + dy)</span></div>
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<span class="s2"></span><br /></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 16</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">Infinite series are expressions such as</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>1 + .5 + .25 + .125 + .0625+....</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">1 + .5 + .25 + .125 + .0625+....+1/2^n = 2 - 1/2^n = 2 - 1/infinity = 2 - 0 = 2</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">continuous functions do not include jumps, or gaps, in the graph of the function</span></div>
<div class="p1">
<span class="s2">the line will not break if it is continuous</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 17</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">the chances of throwing a combination of numbers from 2 dice are as follows:</span></div>
<div class="p1">
<span class="s2">2: 1/36</span></div>
<div class="p1">
<span class="s2">3: 2/36</span></div>
<div class="p1">
<span class="s2">4: 3/36</span></div>
<div class="p1">
<span class="s2">5: 4/36</span></div>
<div class="p1">
<span class="s2">6: 5/36</span></div>
<div class="p1">
<span class="s2">7: 6/36</span></div>
<div class="p1">
<span class="s2">8: 5/36</span></div>
<div class="p1">
<span class="s2">9: 4/36</span></div>
<div class="p1">
<span class="s2">10: 3/36</span></div>
<div class="p1">
<span class="s2">11: 2/36</span></div>
<div class="p1">
<span class="s2">12: 1/36</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s2">This chapter reminds me of using trees in probability, where you can multiple the chances out on the branches to get the end probability for a branch.</span></div>
<div class="p1">
<span class="s2">Using counting measures such as the choosing, means saying (5 choose 2) is the same as the total number of combinations of 2 items from a set of 5, which in this case is 10.</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 19</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">Using two functions of a profit model can be used to find the optimal combination of each model, and this relates to economics (which is my second major)</span></div>
<div class="p2">
<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1"><u>Chapter 20</u></span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s2">If axiomatic set theory is consistent, there exists theorems which can neither be proved nor disproved. There is no constructive procedure which will prove axiomatic set theory to be consistent.</span></div>
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<span class="s2"></span><br /></div>
<div class="p1">
<span class="s1">Final Thoughts</span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<br />
<div class="p1">
<span class="s2"> This book is awesome, and is a tough read. A lot of the ideas are familiar, and relate back to many studies in high school and college. At the same time, they expand on so many subjects. I definitely feel that I will reread parts of the book, especially the last 3 chapters. At some point, I feel that any mathematics student should read this during their time at a university. </span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-11871533490183827362013-07-07T12:43:00.001-07:002013-07-07T12:43:14.199-07:00Week 8 - 2nd Book Review (Communication)<div class="p1">
<u>The Math Book - by Clifford Pickover</u></div>
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<span class="s1">This book was a very enjoyable read, and I like how it didn’t use chapters. Each page is a new topic, and it is more like reading a comprehensive timeline rather than a book itself. I would compare to study/looking down a massive wall at a museum, in that it starts with concepts from thousands of years ago, and adds more to the history of math as it becomes closer to the present. By the halfway point of the book, you are already in the 19th century, so the book really focuses on the modern aspect of mathematics. I like how it does so since what we learn through grade school is more about the developments in mathematics that have come in more recent time. </span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">Another enjoyable aspect is that along with each page being a new topic, there is some sort of image that relates to each topic. It makes it a lighter, and more fun read and allows the reader to think further into each subject by giving them an image to start with. </span></div>
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<br /></div>
<div class="p1">
<span class="s1">This is the Koch Snowflake, and is one of the topics discussed in the book. </span></div>
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<div class="p1">
<span class="s1">Many of the pages have other objectives to them, whether it’s thinking about a paradox, solving a problem, or discussing an abstract physical relation to a subject. It’s a very interactive book.</span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">An interesting subject is the Infinite Monkey Theorem that was first thought of by Emile Borel in 1913. The idea is that if a monkey is typing random keys on a typewriter, then it would have a 1/(93^56) chance of typing a 56 character phrase. (The book uses “In the beginning, God created the heavens and the earth”). If the monkey is typing at a rate of one key/character a second, then it was calculated to take 10^100 seconds to eventually type the phrase. He states, in the original article where this is discussed, that if you have a room of monkeys that type for 10 hours a day, that the monkeys could write entire museums-worth of books by randomization. </span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">Another interesting paradox in the book is the discussion of Hilbert’s Grand Hotel, where there are an infinite number of rooms. Even though the hotel has no vacancy, it is possible to give any number of people a room. This is possible by constantly moving people down a room whenever a room is needed. The hotel is always full, but is always open for more people. </span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">A fascinating topic is the Perfect Magic Tesseract, which involves playing Magic Squares on a four-dimensional cube. The numbers that are used are from 1 to N^4. The magic square involves having an equal sum for all rows, columns, pillars, files, quadragonals, diagonals, and triagonals (“space diagonals of the cubes of the tesseract). John Hendricks that the tesseract cannot be solved in orders below 16, and found that the numbers of order 16 go from 1 to 65,536, and has a magic sum of 534,296. I could not imagine trying to compute that in a rational amount of time. </span></div>
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<span class="s1">The Math Book is a fantastic read for anyone who wants to know about mathematics. Compared to the first book I read, The Book of Modern Math, I saw a similarity in how the books were presented in that they both focus on progressing topics individually. The books ascend through subjects in math by going into detail on each. There is so much to comprehend, but so much to enjoy and have fun with. </span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-12855921791729157252013-07-07T12:31:00.000-07:002013-07-07T12:31:00.061-07:00Week 7 - Paradoxes, Rooms, and Cubes (Doing)<div class="p1">
<span class="s1"><u>Paradoxes, Rooms, and Cubes</u></span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">This week I want to look into more concepts from The Math Book, which is the second book I read for the semester. </span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">The first concept I want to look into more is Newcomb’s Paradox. Here is the paradox: there are two arks. Ark 1 contains $1,000, and Art 2 contains either a spider or the Mona Lisa. There are angels that decide what goes into Art 2, and allegedly they are excellent predictors. Thus, they know what to choose based on what you choose. You are allowed to select either Ark, or both of them. The issue is what you want to choose. You could just choose Ark 1, in which the angels would have predicted you do so, and would have put the Mona Lisa in Ark 2. So would you risk leaving an expensive painting behind? You could just select Ark 2, which the angels would also predict, and they would have put the Mona Lisa in that Ark, but then you’d risk leaving $1,000. Why would anyone risk leaving an extra $1,000? You could select neither, but you’d be an idiot to leave a guaranteed $1,000 behind (and, at least, a pet spider, or, at most, the Mona Lisa). You could also select both, but the angels would predict that you are trying to get the most out of this decision, and place the spider in Ark 2. Now it seems simple, you would probably select just Ark 2 for the Mona Lisa, but the issue comes like this: the angels made their prediction a vast amount of time before you were presented with the choice. This plays serious mind-games with those who make the decision. </span></div>
<div class="p2">
<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">A potential solution would be to use a time-machine, but then free-will would not be possible. It’s also a fact that time-machines don’t exist...yet. After pondering this paradox, I’d take Ark 2 only. It’s similar to the prisoners’ problem in game theory, in which choosing the safer option no matter what the opposer would decide to do. IF the angels predict me choosing only Ark 2, then I will have the Mona Lisa. If they have an inaccurate prediction, then they are liars, and I’ll sock them a good one where it counts. </span></div>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1">Another concept is Tokarsky’s Unilluminable Room. You are in a room of any size, in any shape, with side passages. Every wall is covered in mirrors. If I am in the room with you, and I light a match, will you be able to see it from anywhere in the room? Tokarsky came up with the concept in 1969, and it perplexed mathematicians for decades. Assuming all walls are straight, Tokarsky found that a 24-sided room could be possible in which the light wouldn’t be seen from certain locations.</span></div>
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<span class="s1">Finally, I want to study how to solve a Rubik’s Cube. (I have never tried to actually solve one before).</span></div>
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<span class="s1">It’s easy to divide the cube into sides and attempt to solve them one by one. For one side, make a cross so that the adjacent sides have different colored pieces that extend from the cross. </span></div>
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<span class="s1">Move the cross to the bottom. Solve each corner of the cross side by rotating the far side upward, then rotate the top to the left, and rotate the first side back down.</span></div>
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<a href="http://3.bp.blogspot.com/-p-fLpu_DJWA/UdnAXoO6VmI/AAAAAAAAABw/KJtpsoa476k/s1600/c5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-p-fLpu_DJWA/UdnAXoO6VmI/AAAAAAAAABw/KJtpsoa476k/s1600/c5.png" /></a></div>
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<span class="s1">This completes the first side. </span></div>
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<a href="http://4.bp.blogspot.com/-_QMHu_Zhdas/UdnAYJV3IZI/AAAAAAAAAB8/4HKJfnfxZRY/s1600/c6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-_QMHu_Zhdas/UdnAYJV3IZI/AAAAAAAAAB8/4HKJfnfxZRY/s1600/c6.png" /></a></div>
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<span class="s1">Place the completed side on the bottom. Then follow this algorithm to complete the first two levels on the adjacent sides to the completed side. </span></div>
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<span class="s1">Rotate top level to the left, rotate far right side up, rotate top level to the right, rotate far right side down, rotate top level to the right, rotate front face counterclockwise once, rotate top level to the left, and front face clockwise once.</span></div>
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<a href="http://3.bp.blogspot.com/-GUNxHKBWLYw/UdnAYFxVDxI/AAAAAAAAACQ/cxy4Wwcs9b0/s1600/c7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-GUNxHKBWLYw/UdnAYFxVDxI/AAAAAAAAACQ/cxy4Wwcs9b0/s1600/c7.png" /></a></div>
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<span class="s1">The bottom side should be completed, with the first two levels matching on each adjacent side. </span></div>
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<span class="s1">Next, match color-sharing colors to their sides. Orient the corners by following this algorithm: rotate far right side up, rotate top level to the left, rotate far right side down, rotate top level left, rotate far right side up, rotate top level left two times, rotate right side down, rotate top level left two more times. </span></div>
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<span class="s1">It should look like this:</span></div>
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<a href="http://2.bp.blogspot.com/-ZcdaQvsLvdE/UdnAYVwIMxI/AAAAAAAAACI/NHmLfUA2lws/s1600/c8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-ZcdaQvsLvdE/UdnAYVwIMxI/AAAAAAAAACI/NHmLfUA2lws/s1600/c8.png" /></a></div>
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<span class="s1">Next is to permute the edges. If the cube looks like this: </span></div>
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<a href="http://3.bp.blogspot.com/-zCSCR7dSja8/UdnAY-AoH9I/AAAAAAAAACU/_4h650h_twI/s1600/c9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-zCSCR7dSja8/UdnAY-AoH9I/AAAAAAAAACU/_4h650h_twI/s1600/c9.png" /></a></div>
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<span class="s1">then follow these steps:</span></div>
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<span class="s1">Rotate middle column up, top level right, middle column down, top level right twice, middle column up, top level right, and middle column down.</span></div>
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<span class="s1">For this look:</span></div>
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<a href="http://1.bp.blogspot.com/-5Qf80Jbru-0/UdnBVU5ZD6I/AAAAAAAAACs/arBvMe9Kzok/s1600/c13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-5Qf80Jbru-0/UdnBVU5ZD6I/AAAAAAAAACs/arBvMe9Kzok/s1600/c13.png" /></a></div>
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<span class="s1">Rotate middle column up, top level left, middle column down, top level left twice, middle column up, top level left, and middle column down.</span></div>
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<span class="s1">It should look like this:</span></div>
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<a href="http://1.bp.blogspot.com/-EuIlDAYXCBY/UdnAWRN8UbI/AAAAAAAAACg/RG6OUUqfmCU/s1600/c11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-EuIlDAYXCBY/UdnAWRN8UbI/AAAAAAAAACg/RG6OUUqfmCU/s1600/c11.png" /></a></div>
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<span class="s1">The final series: rotate right side down, middle level left, right side up twice, middle level right twice, right side down, top level left twice, right side up, middle level right twice, right side down twice, middle level left, right side up, top level left twice, and that’s it! Cube solved. </span></div>
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<span class="s1">After trying it a couple times, it started making more sense and became more recognizable. Although, it will take a few more attempts to have it remotely memorized. </span></div>
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<span class="s2">Sources: <a href="http://www.wikihow.com/Solve-a-Rubik's-Cube-(Easy-Move-Notation)"><span class="s3">http://www.wikihow.com/Solve-a-Rubik's-Cube-(Easy-Move-Notation)</span></a></span></div>
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<span class="s3"><a href="http://en.wikipedia.org/wiki/Newcomb's_paradox">http://en.wikipedia.org/wiki/Newcomb's_paradox</a></span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-16385648395815185242013-07-07T12:15:00.000-07:002013-07-07T13:13:02.874-07:00Week 6 - Project Beginnings, and MTG (History) <div class="p1">
<u>Richard Garfield, the start of MTG, and more about the game</u></div>
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<span class="s1">One of my favorite hobbies is playing Magic: The Gathering; a card game that is one the of the first trading card games ever made. Created in 1993 by Richard Garfield, MTG has grown into a popular game that has formats involving real cards, and now has spread to the internet scene in the form of an online version of the game. There are competitive tournaments all over the country on a weekly basis; from small-scale events such as FNM’s (Friday Night Magic) to the Pro Tour and World Championships that happen for each set that is released or every year respectively. There are over twelve million players worldwide, and the game has a major first and secondary markets, sending the game to a high level of popularity. The game involves skills in mathematics, logic, and is even like poker in the sense that gameplay can be determined by bluffing plays or setting traps. There are many different card types, and many different mechanics to the game, with new cards (that have new, and old, abilities) being released every year. There are over 12,000 unique cards, and that number grows every few months with 4 sets being released every year. Three sets are released as a “block” every year, with one core set being released in between blocks each year.</span></div>
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<span class="s2"><u>Richard Garfield</u></span></div>
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<span class="s1">Garfield graduated with a degree in computer mathematics in 1985, and continued on to study combinatorial mathematics. Garfield tried to find a publisher for another game he had designed, but met Peter Adkison of Wizards of the Coast, and they developed the beginning of MTG by wanting to build a game that required a minimal amount of material for a game. MTG went into development while Garfield studied at University of Pennsylvania, and many of the play-testers went to that same school. He got his Ph.D in combinatorial mathematics in 1993, the same year MTG began.Play-testers developed some packs and cards, and Garfield would edit the game appropriately. He joined WotC a year later to develop the game further as a game designer. He still continues to contribute to the game, and has developed many other games since MTG was launched.</span></div>
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<span class="s2"><u>Mathematical Magic</u></span></div>
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<span class="s1">Like so many games that have come since MTG was created, math has become a crucial part to the strategy of play. A positive aspect to learning a game that is complex is that those who handle situations properly, will learn how to solve problems outside of the game. It takes a rational player to understand how the game works. You have to manage your life-total, your mana (land) base, drawing, casting, and attempt to know what your opponent will play before they make a play, or react accordingly to anything. </span></div>
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<span class="s1">In a nutshell, you create a deck that is based on colors of MTG; those are white, blue, black, green, and red. Depending on the format, you choose spells to play as long as you think you can cast them in a game. It’s normal to see two- and three-color decks. You cast a spell by having mana to “tap” for a color, adding that color to a mana pool. You then use the colors in the mana pool to cast a spell for its casting cost. The trick is balancing casting costs with the potential mana-base. If you have access to mana that can tap for different colors (know as “dual lands”), then it becomes easier to play more colors. </span></div>
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<a href="http://1.bp.blogspot.com/-vHPgSB8BVG0/Udm9_0glMfI/AAAAAAAAAAo/05Dr2KIoMEU/s1600/pillar.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-vHPgSB8BVG0/Udm9_0glMfI/AAAAAAAAAAo/05Dr2KIoMEU/s1600/pillar.jpeg" /></a></div>
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<span class="s1">This card costs one red mana to play. (Notice the casting cost in the upper right-hand corner). </span></div>
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<a href="http://2.bp.blogspot.com/-o95LEaBu5Uo/Udm-ErRWkJI/AAAAAAAAAAw/ql3bGuoOmwg/s1600/SF.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-o95LEaBu5Uo/Udm-ErRWkJI/AAAAAAAAAAw/ql3bGuoOmwg/s1600/SF.jpeg" /></a></div>
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<span class="s1">This land can add a red mana or white mana to your pool. The corner-bend symbol you see means the land must be tapped to add a mana, and once it is tapped, it cannot be used again until the next turn. </span></div>
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<span class="s1">Each player takes alternating turns, in which they go through each phase in order:</span></div>
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<li class="li1"><span class="s1">Untap step</span></li>
<li class="li1"><span class="s1">Upkeep effects</span></li>
<li class="li1"><span class="s1">Draw step</span></li>
<li class="li1"><span class="s1">Main phase 1</span></li>
<li class="li1"><span class="s1">Combat phase</span></li>
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<li class="li1"><span class="s1">Declare attackers</span></li>
<li class="li1"><span class="s1">Declare blockers</span></li>
<li class="li1"><span class="s1">Assign damage</span></li>
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<li class="li1"><span class="s1"><span class="Apple-tab-span"> </span> Main phase 2</span></li>
<li class="li1"><span class="s1"><span class="Apple-tab-span"> </span> End step</span></li>
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<span class="s1">Whenever a creature attacks, or land is tapped for mana, or other tapping effects occur for a player, their cards become untapped at the beginning of their turn. Upkeep effects are residual effects that some cards have, and those take place after untapping. The draw phase is simply drawing one card for the turn. The main phase is when creatures, sorceries, enchantments, planeswalkers, lands, and artifacts can be cast. (The other major card type, instants, can be cast anytime that a player has priority). The combat phase is when a player attempts to deal damage to their opponents’ life total. They tap their creatures to show they are attacking. </span></div>
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<a href="http://1.bp.blogspot.com/-IHB-GliNg4E/Udm-J-dcAzI/AAAAAAAAAA4/-bwsdPmA3Cw/s1600/thraggy.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-IHB-GliNg4E/Udm-J-dcAzI/AAAAAAAAAA4/-bwsdPmA3Cw/s1600/thraggy.jpeg" /></a></div>
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<span class="s1">They assign combat damage that is equal to the first number, in the fraction, in the bottom right-hand corner of the card. This creature, Thragtusk, has a power of 5, and a toughness of 3. This means that it assigns 5 points of damage whenever it would deal damage, and requires 3 points of damage to kill. A player declares which creatures they are attacking with, and then the defending player declares which (if any) creature they are blocking with. Attacking creatures that are blocked will deal damage to the blocking creatures, while the blocking creature deals damage to the attacking creature that it has blocked. Unblocked creature deal damage to the defending players’ life-total. The second main phase is another chance to cast the same cards that could be casted in the first phase. After the second phase, the turn ends, and the other player takes their turn in the same fashion. </span></div>
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<span class="s1">Instants can be casted at any time they have priority, but that’s for another paper. Another major concept to the game is the “stack”, which follows a “first-in, last-out” idea. As the same with instants, the stack is better left for later, as it is the most complicating part of the game (in my own opinion). </span></div>
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<span class="s1">The math behind this game lies with casting cards at the right time, so you can imagine that it does not take just a mathematician to play, it requires a thought process that can comprehend proper decision-making in short periods of time. It is definitely a game worth playing, and I would encourage anyone to try it who wants a hobby that is more intellectually stimulating than video games, more addictive than drugs, and better than doing something stupid on a Saturday, like making salsa. </span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-66921649124771278242013-07-07T12:09:00.000-07:002013-07-07T12:09:54.333-07:00Week 5 - Solving Cubic Equations (Doing)<div class="p1">
<span class="s1"><u>Cubic Formula Derivation</u></span></div>
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<span class="s1">The cubic equation is of the form:</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>x^3 + ax^2 + bx + c = 0</span></div>
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<span class="s1">Cardano solved the cubic equation by doing these steps:</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>x = y - a/3</span></div>
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<span class="s1">via the Tchirnhaus transformation, and this simplifies the equation to:</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>y^3 +py + q = 0</span></div>
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<span class="s1">where <span class="Apple-tab-span"> </span></span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>p = b - (a^2)/3</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>and<span class="Apple-tab-span"> </span>q = c - ab/3 + (2a^3)/27</span></div>
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<span class="s1">then substitute y = u - v to get:</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(u-v)^3 + p(u-v) + q = 0</span></div>
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<span class="s1">and this can be written as:</span></div>
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<span class="s1"><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span><span class="Apple-tab-span"> </span>(q - (u^3 - v^3)) + (u+v)(p - 3uv) = 0</span></div>
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<span class="s1">With the two separate terms, we see that q = u^3 - v^3 and q = 3uv, and this solves the equation. </span></div>
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<span class="s1">As for the remaining terms in the equations for q, we can solve for u or v and use substitution to finish the equation.</span></div>
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<span class="s1">These roots are solved using methods such as Newton’s Method. The numbers a, b, c, d, u, v, q, and p are real numbers.</span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-3127609519625112732013-07-07T12:06:00.003-07:002013-07-07T12:06:57.581-07:00Week 4 - Islamic Tessellations (Doing)<div class="p1">
<span class="s1"><u>Islamic Tessellation</u></span></div>
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<span class="s1">This week I will be making an Islamic tessellation.</span></div>
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<span class="s1"> I found a youtube video of how to draw one tessellation, and I tried free-handing it, but it looked terrible. I also tried drawing it to scale on a regular sheet of paper, and I drew it out of whack. I then tried making it in a powerpoint slide, but it became tricky. Now I am back to using my hands, but I got smart and started using straight-edges and circular objects to make my tessellation. This was really fun, and I’ve noticed that an Islamic tessellation is more geometrical, in the sense that there isn’t as much overlap as a typical tessellation, and more clear and normal polygons appear in the patterns. There are also more stars, which is very cool. </span></div>
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<span class="s1">For the tessellation that I made, it definitely was more difficult than I thought. Trying to draw it semi-free handed, it didn’t come out that well, but I also wasn’t able to simplify the tessellation to just the final drawing, and all the construction lines are still visible. It was still enjoyable and cool to try.</span></div>
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<a href="http://2.bp.blogspot.com/-sJEt9o_yCe0/Udm8Ldc4v6I/AAAAAAAAAAY/NP9drCAkPRI/s1600/tess.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="240" src="http://2.bp.blogspot.com/-sJEt9o_yCe0/Udm8Ldc4v6I/AAAAAAAAAAY/NP9drCAkPRI/s320/tess.jpg" width="320" /></a></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-78274330934525801712013-07-07T12:03:00.002-07:002013-07-07T12:03:11.240-07:00Week 3 - What is Algebra (Nature)<div class="p1">
<span class="s1"><u>What is Algebra?</u></span></div>
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<span class="s2">Algebra is one of the many parts of mathematics, and will always be a base set of theories and ideas that support mathematic’s ever-infinitely expanding field. It involves all basic thought on equations and number theory. Balancing, defining, and solving equations and understanding numbers is what everyone achieves by learning about algebra. </span></div>
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<span class="s2">Algebra uses non-numerical values to represent unknowns in equations. The idea behind solving these has moved into other forms such as vectors, and matrices. Algebraic structures include sets, rings, fields, and groups. Algebra contains arithmetic, and simple calculations. Many ideas for graph theory stem from algebra, and many students first learn how to graph from algebraic equations that are linear, cubic, quadratic, etc., and this continues into calculus based graphing and other mathematical forms. </span></div>
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<span class="s2">Algebra differs from other branches of mathematics such as calculus, geometry, discrete mathematics, linear algebra, and in a sense; proof-based mathematics, but algebra has its hand in all of these branches in some way. Algebra is really a simpler-to-understand branch of mathematics, but becomes the foundation for all other branches. It’s easy to that algebra is a part of many other fields of math, but not all that fields create algebra. For instance, algebra is learned before calculus because solving something like a differential equation or calculating an integral cannot be done without knowing how to solve an equation to some extent, and the basics for equation-solving comes from learning algebra. However, you don’t need to know how to take a derivative to know how to do any calculation that is algebraic to its most basic sense. Algebra also came fro arithmetic, whereas other developments in other branches of math came from algebra. </span></div>
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<span class="s2">Algebra is a way to connect relationships in math, and it helps develop overall computational skills to help a student in their development. It’s a logical tool that can be used to analyze situations that do not have to be mathematical. Algebra’s evolving as a mathematical tool has paved the way for other subjects, and no other comes to mind more quickly than statistics. Equations and algorithms are all about what algebra is, and nailing down algebraic concepts have lead to advancements in statistics. The logical side to algebra allows students to make better decisions in any aspect of life. It’s easy to figure out simple problems, but a more complex issue that has multiple factors can be rationalized into an algebraic idea and can be solved more efficiently. While every student should learn about algebra, it takes a certain amount of time to develop the skills relating to the subject, and waiting for an appropriate time to learn it is important. I took my first algebra-based course in 8th grade, while most students in the Davison school district took it in 9th grade, and is a requirement to continue to 9th/10th grade math (which is geometry). While I struggled with the concepts of algebra initially, it became more clear as I worked with it, and it helped me mature into a better problem-solver. This is what algebra is and what it does for those that can take advantage of learning about this valuable subject. </span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-83240540380988169192013-05-19T20:30:00.001-07:002013-07-07T12:01:35.966-07:00Week 2 - What's an Axiom? (Nature) <br />
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<span class="s1">What’s an Axiom?</span></div>
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<span class="s2"><span class="Apple-tab-span"> </span>Based on any sort of mathematical knowledge I have, I imagine that an axiom is simply something to believe in relation to mathematics. When something cannot be proved, but seems to always be true, it is assumed to be true. This is what I have believed to be an axiom. Anything I did know essentially came from studying in a course at Grand Valley, and axioms were heavily used in proofs. </span></div>
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<span class="s2"><span class="Apple-tab-span"> </span>Axioms are actually used universally in logic. They stem from our basic sense of reasoning, and are used in the beginning of logic thoughts or arguments. An axiom is specifically a premise that is accepted as a truth since it is very evident. Axioms came from the Greeks, in part to their development of proofs and logic. Euclid was essential in the development of axioms, and deduced many ideas that developed into axioms that believed to always be true without proof. In Euclidian geometry, there are five basic axioms:</span></div>
<ol class="ol1">
<li class="li3"><span class="s3"></span><span class="s2">Given any two points, they can be joined by exactly one line.</span></li>
<li class="li3"><span class="s3"></span><span class="s2">Given any finite, non-zero length line segment, it can be extended infinitely into<br />
exactly one line.</span></li>
<li class="li3"><span class="s3"></span><span class="s2">Given any line segment, there is exactly one circle with one endpoint of the segment as the center, and with the other endpoint on the circle.</span></li>
<li class="li3"><span class="s3"></span><span class="s2">All right angles are equivalent modulo translation, rotation, and mirroring.</span></li>
<li class="li3"><span class="s3"></span><span class="s2">Given a line l and a point p which is not on l, there is exactly one line that passes through<br />
p but never intersects l</span><span class="s3">.</span></li>
</ol>
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<span class="s2"></span><br /></div>
<div class="p3">
<span class="s2">All other ideas were meant to proven.</span></div>
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<span class="s2"></span><br /></div>
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<span class="s4"><span class="Apple-tab-span"> </span>Most axioms were used the different kinds of sciences when they were developed, but mathematics has altered that in the last 200 years. Now there are 9 axioms by way the </span><span class="s2">Zermelo-Fraenkel set theory. The axioms are used to develop proofs in modern mathematics. They have been part of other sciences such as physics, and the axiom remains as the expanding set of believed truths that continue to develop the sciences.</span><br />
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</span><br />
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<span class="s2"><span class="s1">Works Cited:</span></span></div>
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</span>
<br />
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<span class="s2"><span class="s2"><a href="http://www.mrc.uidaho.edu/~rwells/Critical%20Philosophy%20and%20Mind/Chapter%2023.pdf">http://www.mrc.uidaho.edu/~rwells/Critical%20Philosophy%20and%20Mind/Chapter%2023.pdf</a></span></span></div>
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</span>
<div class="p2">
<span class="s2"><span class="s2"><a href="http://scienceblogs.com/goodmath/2007/03/07/basics-axioms/">http://scienceblogs.com/goodmath/2007/03/07/basics-axioms/</a></span></span></div>
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</span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com2tag:blogger.com,1999:blog-7873166440515569669.post-23023651136925590462013-05-15T23:02:00.001-07:002013-05-15T23:02:20.311-07:00What is Math?<u>What is Math?</u><br />
<u><br /></u>
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<span class="s1"><span class="Apple-tab-span"> </span>Math is the study of numbers, and can be applied to a seemingly infinite number of real-life situations. The creation of numbers, symbols, and systems seemed to be invented in conjunction with various mathematical systems, where these numbers and symbols could be used, in application, to likely make things easier in life. </span></div>
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<span class="s1"></span><br /></div>
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<span class="s1"><span class="Apple-tab-span"> </span>The major discoveries, or developments, in mathematics, in my opinion are:</span></div>
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<span class="s1"></span><br /></div>
<ol class="ol1">
<li class="li1"><span class="s1">Invention of Calculus</span></li>
<li class="li1"><span class="s1">Development of scientific systems</span></li>
<li class="li1"><span class="s1">The basic number model</span></li>
<li class="li1"><span class="s1">Numbers such as pi or e</span></li>
<li class="li1"><span class="s1">The order of operations</span></li>
</ol>
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<span class="s1"></span><br /></div>
<div class="p1">
<span class="s1"><span class="Apple-tab-span"> </span>These are massive in the mathematics world, and many industries couldn’t exist without these fields of mathematics. Calculus gives us derivatives, and rates, and the idea of slopes and many other things. The development of financial algorithms come from calculus and differential equations. The scientific systems lead a large part in research in technology and medicine. Our basic number model is the foundation of everything we know today in mathematics. Presumably, the basic number model could be the set of integers, but could be interpreted as any set of numbers, since the original set of numbers could have been any of the sets of numbers. Number such as pi or e as used so much in mathematics. There are entire fields and ideology based on them. Finally, the order of operations allow us to use any number system or set of numbers to function in a progressing society. They simplified life and made many things easier to grasp, as well as became easy for anyone to use. You would be hard pressed to find someone who didn’t understand, at least, part of the order.</span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com0tag:blogger.com,1999:blog-7873166440515569669.post-6411754005573573112013-05-15T23:01:00.002-07:002013-07-07T12:00:58.472-07:00Week 1 - History of Zero (History)<br />
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<span class="s1"><u>History of 0</u></span></div>
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<span class="s2"><span class="Apple-tab-span"> </span>The history of 0 does not necessarily refer to what we believe zero to always be. We would assume that when we hear “zero”, we would think of something that is actually nothing. “I have zero chance of buying this car”, or “There is zero food left at the buffet....shame”. In history though, dating back to Babylonians before Christ, zero did have its existence, as a measurement. More so, zero represented larger quantities, much like how we have tens and hundreds and thousands and so on. </span></div>
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<span class="s2"><br /></span></div>
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<span class="s2"><span class="Apple-tab-span"> </span>It is impossible to tell where zero came from, and cannot be traced to a single source. The concept of zero has been around for centuries, and is thought to be initially used throughout the Middle East, India, and North Africa in the Arabic numerical system, and brought to Europe by Fibonacci in the 13th century. There is evidence that zero was used by the Babylonians but was not written as a single digit or symbol, but interpreted as such. From there, zero became more recognizable, and became more known as how we would typically think of it: as the average between -1 and 1. </span></div>
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<span class="s2"><br /></span></div>
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<span class="s2"><span class="Apple-tab-span"> </span>Now, we see zero as applicable as any given single item in mathematics. It became the center of origin in coordinate planes. More familiarly, it represents an empty space. Oddly enough, the Greeks were believed to have put a symbol to zero, as their numbers were not represented by digits, but by lines. Thus, with zero space not be able to represented by any physical measure, they used “0” to represent the empty space in their numerical system. </span></div>
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<span class="s2"><br /></span></div>
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<span class="s2"><span class="Apple-tab-span"> </span>To get a rough interpretation of the use of zero by all sorts of history’s cultures and societies, I’d recommend reading: <span class="s3"><a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html">http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html</a></span></span></div>
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<span class="s2"><br /></span></div>
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<span class="s2">Other works cited: </span></div>
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<span class="s1"><a href="http://www.scientificamerican.com/article.cfm?id=history-of-zero">http://www.scientificamerican.com/article.cfm?id=history-of-zero</a></span></div>
holtjefhttp://www.blogger.com/profile/03709581351550309082noreply@blogger.com1