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copied!<p>For a surreal-like decimal expansion, you need a random binary number. <em>Even</em> bits tell you whether to stop or continue, <em>odd</em> bits tell you whether to go right or left on the tree:</p> <pre><code>> 0... => 0.0 [50%] Stop > 100... => -0.5 [<12.5%] Go, Left, Stop > 110... => 0.5 [<12.5%] Go, Right, Stop > 11100... => 0.25 [<3.125%] Go, Right, Go, Left, Stop > 11110... => 0.75 [<3.125%] Go, Right, Go, Right, Stop > 1110100... => 0.125 > 1110110... => 0.375 > 1111100... => 0.625 > 1111110... => 0.875 </code></pre> <p>One way to quickly generate a random binary number is by looking at the decimal digits in math.random() and replace 0-4 with '1' and 5-9 with '1':</p> <ul> <li><p>0.8430419054348022 becomes 1000001010001011 which becomes <code>-0.5</code> </p></li> <li><p>0.5513009827118367 becomes 1100001101001011 which becomes <code>0.25</code> etc</p></li> </ul> <p>Haven't done much lua programming, but in Javascript you can do:</p> <pre><code>Math.random().toString().substring(2).split("").map( function(digit) { return digit >= "5" ? 1 : 0 } ); </code></pre> <p>or true binary expansion:</p> <pre><code>Math.random().toString(2).substring(2) </code></pre> <p>Not sure which is more genuinely "random" -- you'll need to test it.</p> <p>You <em>could</em> generate surreal numbers in this way, but most of the results will be decimals in the form a/2^b, with relatively few integers. On Day 3, only 2 integers are produced (-3 and 3) vs. 6 decimals, on Day 4 it is 2 vs. 14, and on Day n it is 2 vs (2^n-2).</p> <p>If you add two uniform random numbers from <code>math.random()</code>, you get a new distribution which has a "triangle" like distribution (linearly decreasing from the center). Adding 3 or more will get a more 'bell curve' like distribution centered around 0:</p> <pre><code>math.random() + math.random() + math.random() - 1.5 </code></pre> <p>Dividing by a random number will get a truly wild number:</p> <pre><code>A/(math.random()+1e-300) </code></pre> <p>This will return an results between A and (<em>theoretically</em>) A*1e+300, though my tests show that 50% of the time the results are between A and 2*A and about 75% of the time between A and 4*A.</p> <p>Putting them together, we get:</p> <pre><code>round(6*(math.random()+math.random()+math.random() - 1.5)/(math.random()+1e-300)) </code></pre> <p>This has over 70% of the number returned between -9 and 9 with a few big numbers popping up rarely.</p> <p>Note that the average and sum of this distribution will tend to diverge towards a large negative or positive number, because the more times you run it, the more likely it is for a small number in the denominator to cause the number to "blow up" to a large number such as 147,967 or -194,137.</p> <p>See <a href="https://gist.github.com/3957208" rel="nofollow">gist</a> for sample code.</p> <p><a href="http://about.me/joshuaweinstein" rel="nofollow">Josh</a></p>

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