**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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**Tags:** Le Monde, mathematical puzzle, R, sudoku