Le Monde puzzle [#1154]

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The weekly puzzle from Le Monde is another Sudoku challenge:

An n by n grid contains all numbers from 1 till n². Is it possible for fill the grid so that every row and every column has an integer average, for n=5, 7 9?

By sheer random search

`?`=rowSums; `+`=sample 
o=function(n){
x=matrix(+(n^2),n) 
while(any(c(?x,?t(x))%%n))x=x/x*+x 
x}

I found solutions for n=3,4,5, quite easily,

     [,1] [,2] [,3] [,4] [,5]
[1,]   20   15   14   13    3
[2,]   21    4   25    6    9
[3,]    2    1   23   18   11
[4,]   17   12   22   24    5
[5,]   10    8   16   19    7

correction, for n=6 as well

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

but larger values of n require a less frontal attack… Simulated annealing maybe.

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This article is originally published at https://xianblog.wordpress.com
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