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In an nxn table, all integers between 1 and n appear n times. If max denotes the maximum over the numbers of different integers on all rows and columns, what is the minimum value of max when n=7? when n=11?

I tried to solve it by the following R code (in a pre-breakfast mode in my Reykjavik Airbnb flat!):

#pseudoku
n=7
T=10^4
vals=rep(1:n,n)
minmax=n
for (t in 1:T){
psudo=matrix(sample(vals),ncol=n)
maxc=maxr=max(sapply(apply(psudo,1,unique),length))
if (maxc<minmax)
maxr=max(sapply(apply(psudo,2,unique),length))
minmax=min(minmax,max(maxc,maxr))
}

but later realised that (a) the

sapply(apply(psudo,1,unique),length)

failed when all rows or all columns had the same number of unique terms and (b) I did not have to run the whole matrix:

vals=rep(1:n,n)
minmax=n
for (t in 1:T){
psudo=matrix(sample(vals),ncol=n)
maxc=max(length(unique(psudo[1,])),length(unique(psudo[,1])))
i=1
while((i<n)&(maxc<minmax)){
i=i+1
maxc=max(maxc,
length(unique(psudo[i,])),
length(unique(psudo[,i])))}
minmax=min(minmax,maxc)
}

gaining a factor of 3 in the R execution. With this random exploration, the minimum value returned was 2,2,2,3,4,5,5,6,7,8 for n=2,3,4,5,6,7,8,9,10,11. Half-hearted simulating annealing during one of the sessions of AISTATS 2014 did not show any difference…