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# [Numberphile] - The Square-Sum Problem

Posted 6 years ago
 As part of my Numberphile series of posts: here is another one about a recent video called The Square-Sum Problem The question is: if you have the integers 1 through n, can you arrange that list in such a way that every two adjacent ones sum to a square number. As seen in the video and the extra footage. We can easily check this in the Wolfram Language: Let's see which number can pair up to a pair: SquareEdges[n_Integer?Positive]:=Reap[Do[If[IntegerQ[Sqrt[i+j]],Sow[{i,j}]],{i,n-1},{j,i+1,n}]][[2,1]]  Now let's try for 15, as in the main video: n = 15; poss = SquareEdges[n]; gr = Graph[TwoWayRule @@@ poss, VertexLabels -> Automatic]; path = FindHamiltonianPath[gr, PerformanceGoal :> "Speed"] HighlightGraph[gr, BlockMap[Rule @@ # &, path, 2, 1]]  giving: {9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8}  In the extra footage, it is revealed that they found the solution for up to n=299. Can we do better? Yes we can! Changing n to 300 in the above code and rerunning gives us the solution in 0.28 sec on my laptop. {289,35,65,259,30,294,67,257,32,292,69,100,44,125,71,154,135,189,211,113,248,8,281,119,205,195,166,158,283,6,250,191,133,156,285,4,252,277,12,244,117,207,193,168,273,16,240,160,164,236,20,269,131,94,230,59,197,92,232,57,199,90,234,22,267,217,224,137,152,73,123,46,150,75,121,48,148,77,179,110,214,270,19,237,163,161,239,17,272,128,41,103,297,27,262,62,227,97,99,190,210,114,175,50,146,79,177,112,212,188,253,3,286,155,134,266,23,233,91,198,58,231,93,196,60,229,95,130,159,165,276,13,243,118,206,194,167,274,15,241,288,1,255,186,138,223,218,143,181,108,88,201,55,170,86,203,53,172,84,37,107,182,142,299,25,264,220,221,140,184,216,225,64,36,85,171,54,202,87,169,56,200,89,235,21,268,132,157,284,5,251,278,11,245,116,208,192,249,7,282,247,9,280,204,120,136,153,72,124,45,151,74,122,47,149,76,180,109,215,185,139,222,219,265,24,300,141,183,106,38,83,173,52,144,81,40,104,296,28,261,63,226,98,127,42,102,298,26,263,61,228,96,129,271,18,238,162,279,10,246,115,209,275,14,242,287,2,254,187,213,111,178,78,147,49,176,80,145,51,174,82,39,105,295,29,260,101,43,126,70,291,33,256,68,293,31,258,66,34,290} and a completely mess of a graph: Can we go beyond? Let's optimize a code a bit, and let's find the solution for larger n: Let's store our intermediate results in the association db: SetDirectory[NotebookDirectory[]]; $HistoryLength=1; db=If[FileExistsQ["squaresumdb.mx"], Import["squaresumdb.mx"] , <||> ];  And now our main code: ClearAll[SquareEdges,SquareEdges2,CheckSol,TryToFind] SquareEdges[n_Integer?Positive]:=Reap[Do[If[IntegerQ[Sqrt[i+j]],Sow[{i,j}]],{i,n-1},{j,i+1,n}]][[2,1]] SquareEdges2[n_Integer]:=Module[{tmp}, tmp=Table[ {i,#}&/@(Range[Ceiling[Sqrt[2 i]],Floor[Sqrt[i+n]]]^2-i) , {i,1,n-1} ]; tmp=Join@@tmp; Select[tmp,Less@@#&] ] CheckSol[l_List]:=Sort[l]===Range[Length[l]]\[And](And@@BlockMap[IntegerQ@*Sqrt@*Total,l,2,1]) TryToFind[n_Integer?Positive]:=Module[{edges,out}, If[!KeyExistsQ[db,n], edges=SquareEdges2[n]; If[Union[Flatten[edges]]===Range[n], edges=TwoWayRule@@@edges; edges=RandomSample[edges]; Do[ out=TimeConstrained[FindHamiltonianPath[Graph[edges],PerformanceGoal:>"Speed"],5+i,$Failed]; If[out=!=\$Failed, If[Length[out]==0, Print[Style["No solution for ",Red],n]; , status=Row[{"Found solution for ",n,":",i}]; ]; AssociateTo[db,n->out]; Break[] ]; Print["Failed ",n,":",i]; edges=RandomSample[edges]; , {i,5} ] , Print["Edges are not connected for ",n]; AssociateTo[db,n->{}] ] ] ]  Let's scan the first 1000: Dynamic[status] status = ""; Do[TryToFind[k], {k, 3, 1000}] Export["squaresumdb.mx", db];  Note that if finding the Hamiltonian path takes too long I mix the edges and try again, sometimes, seemingly random, it then finds the solution quickly. I can tell you now that all of them have a solution. In fact I went up to larger numbers and found that all up to 2667 have a solution, and possibly beyond. I attached the notebook and the solutions in form of a mx file. Attachments:
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Posted 5 years ago
 Recently I designed a surprisingly fast algorithm, which can generate all solutions to any N<=40000 in seconds. I've put my code on GitHub now, along with a clear (hopefully) explanation. This may be a good supplement for the best solution so far, which contains an enormous precalculated table.
Posted 6 years ago
 Huh. I just noticed that the sums for n = 16 have a funny pattern... The solution for this is {16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8}; however, if we take the sequence of "in-between" square sums, we have:{25, 16, 9, 16, 25, 16, 9, 16, 25, 16, 9, 16, 25, 16, 9}I'm sure that this is just coincidence.Otto Felix: As of the moment, I am unsure of how to solve this problem. Hopefully some smart maths students can see this thread. Until then, I'll see if I can find anything else interesting about the problem...Also, I am interested if there are any examples of multiple solutions for the square-sum sequences for the nth case. For anyone computing out there, this may be interesting to pursue...
Posted 6 years ago
 Turns out someone has already solved it on Mersenne forum, just search square sum problem on the forum and there is a thread. The proof is a complicated to completely understand, but not to hard to fathom, and a fun solution.
Posted 6 years ago
 - Congratulations! This post is now a Staff Pick as distinguished by a badge on your profile! Thank you, keep it coming!
Posted 6 years ago
 Does anyone have any ideas on how I would go about mathematically proving that it works for n>24?
Posted 6 years ago
 Example: for n=2500, the solution is: 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
Posted 6 years ago
 I asked the same question 2 years ago :) http://community.wolfram.com/groups/-/m/t/523917
Posted 6 years ago
 I didn't see that topic! Very similar code! Thanks for sharing.