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<title>Nim</title>
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<meta name="author" content="Kostas Giannakakis">
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<h1>Nim</h1>
<h2>The Game</h2>
<p>Nim is a strategy game, where two players take turns removing objects from distinct heaps.
A player can remove any number of items, as long as they all come from the same heap and they
appear in consecutive places. The player who removes the last item from all the heaps wins.</p>
<h2>Strategy</h2>
<p>Depending on the arrange of items an algorithm exists that determines, which player is going to
win. This algorithm is better presented with an example. Consider of arrange of three heaps, one
of size 1, one of size 2 and one of size 3.</p>
<div align="center">
<table class="nim">
<tr><td> </td><td>|</td><td> </td></tr>
<tr class="odd"><td>|</td><td>|</td><td> </td></tr>
<tr><td>|</td><td>|</td><td>|</td></tr>
</table>
</div>
<p>This arrange can be represented as follows:</p>
<div align="center">
<table>
<tr><td>1 = </td><td>001<sub>2</sub></td></tr>
<tr><td>2 = </td><td>010<sub>2</sub></td></tr>
<tr><td>3 = </td><td>011<sub>2</sub></td></tr>
<tr><td> </td><td>---</td></tr>
<tr><td><em class="sum">Sum:</em> </td><td>000</td></tr>
</table>
</div>
<p>For each row we count the number of elements. We convert the number to its binary representation.
A binary number consists of 1s and 0s. In each column we count the number of 1s. If the result is
an odd number, then in the corresponding column at the <em class="sum">Sum</em> we get an 1. If it
is an even number, we get a 0.</p>
<p>If the <em class="sum">Sum</em> is all zeros, then the player who plays first loses. On the other hand,
if there is even a single 1 in the <em class="sum">Sum</em>, then the first player can always find a
move that it will convert the arrange of items into a form that has a zero <em class="sum">Sum</em> and
win.</p>
<p>When you play, it is difficult to convert into binary and calculate the sum quickly. Even if you do so,
you still need to find the move that it will lead to a zero sum arrange and this is not easy either. In
order to play quickly, you can follow a small set of rules:</p>
<ul>
<li>An arrange of 1, 2, 3 loses.</li>
<li>An arrange of 1, 3, 5 wins (convert it to 1, 2, 3).</li>
<li>An arrange, where each number appears an even number of times loses (e.g 1,1,2,2 and 4,4).
You can quickly draw out of the picture duplicating rows.</li>
<li>A combination of a winning and a losing arrange wins.</li>
</ul>
<h2>Game modes</h2>
<p><em class="appname">NimChallenge</em> offers three different game modes. In <em>Classic</em> mode
the original arrange of items is used. The player can select whether to play first or not, but in
order to win the player must start first. You can try this mode to practice and follow computer's
moves.</p>
<p>In <em>Variations</em> mode different arranges of items are used. The player selects, if she wants to
play first or not. This is a crucial decision and the specific arrange of items must be taken into
account. If the player wins, she can continue playing another game. Every time the arrange of items
is getting more complicated. In this mode a score is kept. The player must play as quickly as possible
in order to win more points.</p>
<p>In <em>Challenge</em> mode you play against time. You only have a limited amount of time to find the
best move. If the time expires, you lose. In this mode also a score is kept and faster moves win
more points.</p>
<h2>Settings</h2>
<p>The following aspects of the game can be configured:</p>
<ul>
<li><em>Single Touch:</em> If checked each move needs not to be confirmed and it is instantly played.
This is quicker, but doesn't give a chance to correct an error.</li>
<li><em>Theme: </em> Selects the appearance of the board.</li>
<li><em>Last wins: </em> Selects if the game will be played as a normal (checked) or misère game (unchecked).
In a normal game the player who removes the last element wins. In a misère game the last player loses.</li>
</ul>
<p>The above settings affect all game modes.</p>
<h2>More Nim</h2>
<p>Play more Nim here. Select the pegs by
clicking on them, when you are finished press the button 'Move' to remove them.</p>
<div align="center">
<script language="JavaScript">
//create the pix array
var pix = new Array();
var boardTable = new Array (1,0,0,0,0,
1,1,0,0,0,
1,1,1,0,0,
1,1,1,1,0,
1,1,1,1,1);
pix[0] = new Image();
pix[0].src = 'images/peg_empty.png';
pix[1] = new Image();
pix[1].src = 'images/peg.png';
pix[2] = new Image();
pix[2].src = 'images/peg_sel.png';
pix[3] = new Image();
pix[3].src = 'images/peg_sel_.png';
pix[4] = new Image();
pix[4].src = 'images/frame_v.png';
pix[5] = new Image();
pix[5].src = 'images/frame_h.png';
pix[6] = new Image();
pix[6].src = 'images/frame_c1.png';
pix[7] = new Image();
pix[7].src = 'images/frame_c2.png';
pix[8] = new Image();
pix[8].src = 'images/frame_c3.png';
pix[9] = new Image();
pix[9].src = 'images/frame_c4.png';
//Place this script wherever you want your calendar
//The first argument must match the var name
var thisCandles = new candles('thisCandles', pix, boardTable, false, 5, 5);
//thisCandles.computerTurn = false;
document.write(thisCandles.write());
</script>
<br>
<form name="computerMove" >
<input type="button" value="Move" onClick="thisCandles.move(thisCandles)">
<input type="button" value="New game" onClick="thisCandles.newGame(thisCandles, pc_first.checked)">
<br><br>
<input type="checkbox" name="pc_first" value="0">Computer first
</form>
</div>
</div>
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