Le Monde puzzle [#932]

A 12×8 grid contains the first 96 integers, as in R matrix(1:96,ncol=12). If one picks 24 of those integers including 3 for each row and 2 for each column, what are the extreme values of the sum of the selected integers?

I obviously rephrased quite strongly the question (and possibly changed it!). Before rushing to the R simulation of a random collection of 24 such integers, I pondered how this sum could vary among random samples since there were the same terms in all samples. More clearly, using the 10×10 grid instead as a basis for reasoning, picking e.g. 20 integers with 2 per row and 2 per colum for all rows and columns, we end up with 2 copies of every integer between 0 and 9 and 2 copies of every decimal between 0 and 90. Random simulation confirms this reasoning:

```grid=matrix(1:96,ncol=12)
#pick a subset at random
coleft=sample(rep(1:12,2))
sub7=grid[1,coleft[1:3]]
coleft=coleft[-(1:3)]
for (i in 2:8){
sub7=c(sub7,grid[i,coleft[1:3]])
coleft=coleft[-(1:3)]}
resl=sum(sub7)
```

since repeated call to the above keeps returning the same value, 1164, which is also

```> sum(3*(0:7))*12+2*sum(1:12)
[1] 1164
```