Xi'an's Og

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:

[sourcecode language=”r” gutter=”false”]
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)
[/sourcecode]

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

[sourcecode language=”r” gutter=”false”]
> sum(3*(0:7))*12+2*sum(1:12)
[1] 1164
[/sourcecode]

This entry was posted on October 15, 2015 at 12:15 am and is filed under Books, Kids, Statistics, University life with tags grid, Le Monde, mathematical puzzle, permutation, R. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.