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<blockquote data-quote="CRGreathouse" data-source="post: 2174225" data-attributes="member: 474">Interesting.&nbsp; I wrote a program to calculate the answer using Charwoman Gene's specs (that is, using only integer arithmetic) and I have to admit I came to a very different answer (off by about 3).&nbsp; I wonder if I made a mistake in my coding, since I doubt roundoff from your method is nearly that extreme.This program is written in C# and runs in just a few seconds.&nbsp; It takes 16^6 loops in the main section to finish.&nbsp; Instead of calculating point buy as it goes, it stores the chance of getting each [code]int[] ways = new int[19];long[] stat = new long[19];int a,b,s,d,c,i,w,ch;String o = &quot;&quot;;long total = 0, totalPoints1 = 0, totalPoints2 = 0;for (i = 3; i &lt; 19; i++)&nbsp; &nbsp; stat[i] = 0;for (i = 3; i &lt; 19; i++)&nbsp; &nbsp; ways[i] = 0;for (a = 1; a &lt; 7; a++)&nbsp; &nbsp; for (b = 1; b &lt; 7; b++)&nbsp; &nbsp; &nbsp; &nbsp; for (c = 1; c &lt; 7; c++)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for (d = 1; d &lt; 7; d++)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ways[a+b+c+d-Math.Min(a,Math.Min(b,Math.Min(c,d)))]++;for (s = 3; s &lt; 19; s++)&nbsp; &nbsp; for (d = 3; d &lt; 19; d++)&nbsp; &nbsp; &nbsp; &nbsp; for (c = 3; c &lt; 19; c++)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for (i = 3; i &lt; 19; i++)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for (w = 3; w &lt; 19; w++)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for (ch = 3; ch &lt; 19; ch++) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; {&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if (s &lt; 14 &amp;&amp; d &lt; 14 &amp;&amp; c &lt; 14 &amp;&amp; i &lt; 14 &amp;&amp; w &lt; 14 &amp;&amp; ch &lt; 14)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; break;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if (m(s) + m(d) + m(c) + m(i) + m(w) + m(ch) &lt;= 0)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; break;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; stat[s] += ways[s];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; stat[d] += ways[d];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; stat[c] += ways[c];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; stat[i] += ways[i];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; stat[w] += ways[w];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; stat[ch] += ways[ch];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; }for (i = 3; i &lt; 19; i++) {&nbsp; &nbsp; o += &quot;Ways to get a &quot; + i + &quot;: &quot; + stat[i] + &quot;\n&quot;;&nbsp; &nbsp; total += stat[i];&nbsp; &nbsp; totalPoints1 += pb1(i) * stat[i];&nbsp; &nbsp; totalPoints2 += pb2(i) * stat[i];}o += &quot;\nAverage point buy: \n&quot;;o += &quot;0 method:&nbsp; &nbsp; &nbsp;&nbsp; &quot; + totalPoints1 + &quot;/&quot; + total + &quot; = &quot; + (6.0 * totalPoints1 / total) + &quot;\n&quot;;o += &quot;1-for-1 method: &quot; + totalPoints2 + &quot;/&quot; + total + &quot; = &quot; + (6.0 * totalPoints2 / total) + &quot;\n&quot;;return o;[/code]The program's output:[code]Ways to get a 3: 856665Ways to get a 4: 3426660Ways to get a 5: 11088250Ways to get a 6: 23285325Ways to get a 7: 54198070Ways to get a 8: 88428430Ways to get a 9: 164239075Ways to get a 10: 220188650Ways to get a 11: 267114100Ways to get a 12: 372160335Ways to get a 13: 383302860Ways to get a 14: 432541600Ways to get a 15: 354143435Ways to get a 16: 299238190Ways to get a 17: 171902790Ways to get a 18: 76717725Average point buy: 0 method:&nbsp; &nbsp; &nbsp;&nbsp; 16542089730/2922832160 = 33.95765919723561-for-1 method: 16694113165/2922832160 = 34.2697334321106[/code]</blockquote>

[QUOTE="CRGreathouse, post: 2174225, member: 474"] Interesting. I wrote a program to calculate the answer using Charwoman Gene's specs (that is, using only integer arithmetic) and I have to admit I came to a very different answer (off by about 3). I wonder if I made a mistake in my coding, since I doubt roundoff from your method is nearly that extreme. This program is written in C# and runs in just a few seconds. It takes 16^6 loops in the main section to finish. Instead of calculating point buy as it goes, it stores the chance of getting each [code]int[] ways = new int[19]; long[] stat = new long[19]; int a,b,s,d,c,i,w,ch; String o = ""; long total = 0, totalPoints1 = 0, totalPoints2 = 0; for (i = 3; i < 19; i++) stat[i] = 0; for (i = 3; i < 19; i++) ways[i] = 0; for (a = 1; a < 7; a++) for (b = 1; b < 7; b++) for (c = 1; c < 7; c++) for (d = 1; d < 7; d++) ways[a+b+c+d-Math.Min(a,Math.Min(b,Math.Min(c,d)))]++; for (s = 3; s < 19; s++) for (d = 3; d < 19; d++) for (c = 3; c < 19; c++) for (i = 3; i < 19; i++) for (w = 3; w < 19; w++) for (ch = 3; ch < 19; ch++) { if (s < 14 && d < 14 && c < 14 && i < 14 && w < 14 && ch < 14) break; if (m(s) + m(d) + m(c) + m(i) + m(w) + m(ch) <= 0) break; stat[s] += ways[s]; stat[d] += ways[d]; stat[c] += ways[c]; stat[i] += ways[i]; stat[w] += ways[w]; stat[ch] += ways[ch]; } for (i = 3; i < 19; i++) { o += "Ways to get a " + i + ": " + stat[i] + "\n"; total += stat[i]; totalPoints1 += pb1(i) * stat[i]; totalPoints2 += pb2(i) * stat[i]; } o += "\nAverage point buy: \n"; o += "0 method: " + totalPoints1 + "/" + total + " = " + (6.0 * totalPoints1 / total) + "\n"; o += "1-for-1 method: " + totalPoints2 + "/" + total + " = " + (6.0 * totalPoints2 / total) + "\n"; return o;[/code] The program's output: [code]Ways to get a 3: 856665 Ways to get a 4: 3426660 Ways to get a 5: 11088250 Ways to get a 6: 23285325 Ways to get a 7: 54198070 Ways to get a 8: 88428430 Ways to get a 9: 164239075 Ways to get a 10: 220188650 Ways to get a 11: 267114100 Ways to get a 12: 372160335 Ways to get a 13: 383302860 Ways to get a 14: 432541600 Ways to get a 15: 354143435 Ways to get a 16: 299238190 Ways to get a 17: 171902790 Ways to get a 18: 76717725 Average point buy: 0 method: 16542089730/2922832160 = 33.9576591972356 1-for-1 method: 16694113165/2922832160 = 34.2697334321106[/code] [/QUOTE]

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