## Le Monde puzzle [#817]

**T**he weekly Le Monde puzzle is (again) a permutation problem that can be rephrased as follows:

Find

wheredenotes the set of permutations on {0,…,10} andis defined modulo 11 [to turn {0,...,10} into a torus]. Same question for

and for

**T**his is rather straightforward to code if one adopts a brute-force approach::

perminmax=function(T=10^3){ mins=sums=rep(500,3) permin=matrix(0:10,ncol=11,nrow=3,byrow=TRUE) for (t in 1:T){ perms=sample(0:10) adde=perms+perms[c(11,1:10)] sums[1]=max(adde) adde=adde+perms[c(10,11,1:9)] sums[2]=max(adde) adde=adde+perms[c(9:11,1:8)]+perms[c(8:11,1:7)] sums[3]=max(adde) for (j in 1:3) if (sums[j]<mins[j]){ mins[j]=sums[j];permin[j,]=perms} } print(mins) print(permin) }

(where I imposed the first term to be zero because of the invariance by permutation), getting the solutions

> perminmax(10^5) [1] 11 17 28 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [1,] 0 10 1 6 5 4 7 3 8 2 9 [2,] 0 10 4 3 5 1 9 6 2 8 7 [3,] 0 2 9 6 7 3 1 4 10 8 5

for 2, 3, and 5 terms. (Since 10! is a mere 3.6 million, I checked with an exhaustive search, using the function permutation from the gtools package.)

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