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Another Sudoku-like puzzle from the weekend edition of Le Monde. The object it starts with is a 9×9 table where each entry is an integer and where neighbours take adjacent values. (Neighbours are defined as north, west, south and east of an entry.) The question is about whether or not it is possible to find a table such that the sum of the 81 entries is 99. The problem is equivariant under location change, namely if each entry is shifted by a, the constraints are still satisfied but the sum is modified by 81a. If, for instance, a is the (5,5) entry, the overall sum is 81a plus an even number between -360 and +360. Looking at the generation of such tables, I wrote the simple-minded and lengthy  R code that follows:

```#Generating an acceptable grid
#where neighbours in space must be neighbours in value

grid=matrix(0,9,9)
neigh=grid*0
#start from the centre
neigh[5,5]=1

for (t in 1:8){

neigh[neigh==1]=2

for (i in 1:9){
for (j in 1:9){

if (neigh[i,j]==0){

vois=NULL
if ((j>1)&&(neigh[i,j-(j>1)]==2)){vois=rbind(vois,c(i,j-1))}
if ((j<9)&&(neigh[i,j+(j<9)]==2)){vois=rbind(vois,c(i,j+1))}
if ((i<9)&&(neigh[i+(i<9),j]==2)){vois=rbind(vois,c(i+1,j))}  if ((j>1)&&(neigh[i-(i>1),j]==2)){vois=rbind(vois,c(i-1,j))}

neigh[i,j]=1-is.null(vois)
}

if (neigh[i,j]==1){

if (dim(vois)[1]==1){

grid[i,j]=grid[vois]+sample(c(-1,1),1)}else{

if (min(grid[vois])!=max(grid[vois])){
grid[i,j]=round(mean(grid[vois]))}else{
grid[i,j]=grid[vois[1,1],vois[1,2]]+sample(c(-1,1),1)}
}
}}
}}
```

which provides a random grid centred at zero and satisfying the assumptions. Here is one instance for grid

```> grid
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,]    1    0    1    2    1    0   -1    0   -1
[2,]    0   -1    0    1    0   -1    0    1    0
[3,]    1    0   -1    0    1    0    1    2    1
[4,]    2    1    0    1    2    1    2    1    0
[5,]    1    2    1    0    1    0    1    2    1
[6,]    2    1    0   -1    0    1    0    1    2
[7,]   -1    0   -1    0   -1    0   -1    0    1
[8,]    0    1    0    1    0    1    0   -1    0
[9,]    1    0   -1    0   -1    0   -1    0    1
```

A few thousand simulations show that achieving 99[-81] or 99[+81] (i.e. when centred at 1 or -1) is indeed possible. Here is a grid leading to a total sum of 99:

```       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,]    1    0   -1    0    1    0    1    2    3
[2,]    2    1    0   -1    0    1    2    1    2
[3,]    3    2    1    0    1    2    3    2    3
[4,]    4    3    2    1    2    3    2    1    2
[5,]    3    2    1    2    1    2    1    0    1
[6,]    0    1    2    1    0    1    0    1    0
[7,]    1    0    1    2    1    2    1    0    1
[8,]    0   -1    0    1    2    3    2    1    0
[9,]   -1    0    1    2    1    2    3    2    1
```

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