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copied!<p>OK, this is not elegant or fast, and it's buggy, but it works (sometimes). It uses a <a href="http://en.wikipedia.org/wiki/Monte_Carlo_method" rel="nofollow noreferrer">monte carlo</a> method, implementing the metropolis algorithm for a weight function that I (arbitrarily) selected just to see if this would work. This was some time ago for a similar problem; I suppose my mathematica skills have improved as it looks ugly now, but I have no time to fix it at the moment.</p> <p>Execute this (it looks more reasonable when you paste it into a notebook):</p> <pre><code>ClearAll[swap]; swap[lst_, {p1_, p2_}] := ReplacePart[ lst, {p1 \[Rule] lst\[LeftDoubleBracket]p2\[RightDoubleBracket], p2 \[Rule] lst\[LeftDoubleBracket]p1\[RightDoubleBracket]}] ClearAll[evalops]; (*first element of opslst is Identity*) evalops[opslst_, ord_, nums_] := Module[{curval}, curval = First@nums; Do[curval = opslst\[LeftDoubleBracket]p\[RightDoubleBracket][curval, nums\[LeftDoubleBracket]ord\[LeftDoubleBracket]p\ \[RightDoubleBracket]\[RightDoubleBracket]], {p, 2, Length@nums}]; curval] ClearAll[randomizeOrder]; randomizeOrder[ordlst_] := swap[ordlst, RandomInteger[{1, Length@ordlst}, 2]] ClearAll[randomizeOps]; (*never touch the first element*) randomizeOps[oplst_, allowedOps_] := ReplacePart[ oplst, {RandomInteger[{2, Length@oplst}] \[Rule] RandomChoice[ops]}] ClearAll[takeMCstep]; takeMCstep[goal_, opslst_, ord_, nums_, allowedops_] := Module[{curres, newres, newops, neword, p}, curres = evalops[opslst, ord, nums]; newops = randomizeOps[opslst, allowedops]; neword = randomizeOrder[ord]; newres = evalops[newops, neword, nums]; Switch[Abs[newres - goal], 0, {newops, neword}, _, (p = Abs[curres - goal]/Abs[newres - goal]; If[RandomReal[] < p, {newops, neword}, {opslst, ord}])]] </code></pre> <p>then to solve your actual problem, do </p> <pre><code>ops = {Times, Plus, Subtract, Divide} nums = {25, 50, 75, 100, 3, 6} ord = Range[Length@nums] (*the first element is identity to simplify the logic later*) oplist = {Identity}~Join~RandomChoice[ops, Length@nums - 1] out = NestList[ takeMCstep[ 99, #\[LeftDoubleBracket]1\[RightDoubleBracket], #\ \[LeftDoubleBracket]2\[RightDoubleBracket], nums, ops] &, {oplist, ord}, 10000] </code></pre> <p>and then to see that it worked,</p> <pre><code>ev = Map[evalops[#\[LeftDoubleBracket]1\[RightDoubleBracket], #\ \[LeftDoubleBracket]2\[RightDoubleBracket], nums] &, out]; ev // Last // N ev // ListPlot[#, PlotMarkers \[Rule] None] & </code></pre> <p>giving</p> <p><img src="https://i.stack.imgur.com/GXiMl.png" alt="enter image description here"></p> <p>thus, it obtained the correct order of operators and numbers after around 2000 tries.</p> <p>As I said, it's ugly, inefficient, and badly programmed as it was a quick-and-dirty adaptation of a quick-and-dirty hack. If you're interested I can clean up and explain the code.</p>

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