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**L**e Monde puzzle of last weekend was about sudoku-like matrices.

Consider an (n,n) matrix containing the integers from 1 to n². The matrix is “friendly” if the set of the sums of the rows is equal to the set of the sum of the columns. Find examples for n=4,5,6. Why is there no friendly matrix when n=9?

**C**hecking for small n’s seems easy enough:

friend=function(n){ s=1 while (s>0){ A=matrix(sample(1:n^2),ncol=n) s=sum(abs(sort(apply(A,1,sum))- sort(apply(A,2,sum))))} A }

**F**or instance, running

> friend(3) [,1] [,2] [,3] [1,] 8 4 2 [2,] 1 9 5 [3,] 6 3 7 > friend(4) [,1] [,2] [,3] [,4] [1,] 14 10 11 6 [2,] 13 3 12 8 [3,] 4 5 16 15 [4,] 9 1 2 7 > friend(5) [,1] [,2] [,3] [,4] [,5] [1,] 17 14 10 16 5 [2,] 8 19 7 15 22 [3,] 2 3 25 13 6 [4,] 11 12 18 1 21 [5,] 24 23 20 4 9 >friend(6) [,1] [,2] [,3] [,4] [,5] [,6] [1,] 14 4 36 30 10 18 [2,] 7 16 24 27 32 11 [3,] 21 25 12 5 22 17 [4,] 26 20 6 31 19 34 [5,] 3 35 1 28 9 29 [6,] 23 2 33 15 13 8

produces right answers. But the case n=6 proved itself almost too hard for brute-force handling!!! I have no time to devise a simulated-annealing code to speed up the resolution so this will have to wait till the next weekend edition of Le Monde. As to why n=9 does not enjoy a solution… *(When n=2 there is no solution, as proven by the brute force exhibition of all cases.)*

Filed under: R Tagged: Le Monde, mathematical puzzle, R, sudoku

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