## Le Monde puzzle [#932]

**A** Sudoku-like Le Monde mathematical puzzle:

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

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