Le Monde puzzle [#853]

Yet another one of those Le Monde mathematical puzzles which wording is confusing (or at least annoying) to me:

A political party has 11 commissions, to which belong some of the 13 members of the central committee. A token is given to each member for each commission whom he or she belongs. Two different members cannot share more than one common commission. How many tokens at most? Same question if the president belongs to five commissions.

I just dislike the “story” around the combinatoric problem. Given 13 sets and 11 letters, how many letters can one allocate to each set so that each pair of sets share at most one letter? While waiting for my prosthesis this afternoon, I thought of a purely random search, using the following R code:

n=11
m=13
best=0
for (t in 1:10^3){
  token=matrix(-1,ncol=n,nrow=m);diag(token)=1
  #fill at random: -1 unchecked, 1 allocated, 0 prohibited
  while (sum(token<0)>0){
    res=TRUE
    while (res){
      if (sum(token<0)==1){
          dex=(1:(n*m))[token[(1:(n*m))]<0]}else{
          dex=sample((1:(n*m))[token[(1:(n*m))]<0],1)}
      i=dex%%m+m*(dex%%m==0)
      j=trunc(dex/(1.01*m))+1
      res=FALSE
      for (k in (1:m)[token[,j]==1])
         res=res||(max(token[i,-j]+token[k,-j])==2)
      token[dex]=0
      if (sum(token<0)==0) res=FALSE
      }
    token[dex]=1*(sum(token<0)>0)
    for (k in (1:m)[-i])
      if (max(token[i,-j]+token[k,-j])==2) token[k,j]=0
    image(1:13,1:11,token)
    }
  if (sum(token[token==1])>best) bestoken=token
  best=max(best,sum(token))
}

which led to the value of 43 after that many iterations. Longer runs on faster machines at Paris-Dauphine did not change the output but, as usual with brute force simulation, the true solution may be such an extreme that it is extremely unlikely to happen… I then tried a standard simulated annealing exploration, but could not find an higher value. Except once, leading to the value 44. Here is the corresponding allocation of letters (committees) to sets (individuals) for that solution.In this Feb. 5, 2014, issue of Le Monde Science&Médecine leaflet, a review of (my Warwick colleague) Ian Stewart’s 17 equations that changed the World, which must have been recently translated in French (with no criticism of the less compelling chapter on Black-Scholes, and a confusion of the “bell curve” with statistics). Yet another tribune of Marco Zito about the generalisation of Richard Feynman’s diagrams to a solid called the Amplituhedron. (Not making much sense as exposed!)