Dear all, the French newspaper Le Monde has a weekly math-oriented puzzle. In number 1021 the authors (Elisabeth Busser, Gilles Cohen, Jean-Louis Legrand) have not been able to convince readers of their solution to their puzzle, and seem not to be too sure that their solution is correct. The authors have invited everyone to provide a convincing solution - which I have not been able to do. I thus turn to the collective intelligence of the math-fun list! The puzzle reads as follows (my translation, simplifying a bit from http://www.affairedelogique.com/espace_probleme.php?corps=probleme&num=1021 ): In a democratically advanced country, elections to the municipal council are run in a particular way. The council has a pre-determined size. Once the number of candidates is known, it is decided for how many candidates each voter has to vote [all votes have to be cast; it is not allowed to vote twice for the same candidate]. The number of votes is set so that the composition of the elected council (to be selected by an algorithm based on all the votes) is such that every voter has at least one of the candidates he voted for elected in the council, no matter what the votes where. First problem: in Alice's small city, 10 council members had to be elected. Each voter could vote for 10 candidates, out of a list of 25. What is the largest possible number of voters? Second problem: in Bob's medium-sized city, there were 55,555 voters and 33 candidates. By some coincidence, as in Alice's case, the number of votes that each voter can cast is equal to the size of the council. What is, at the minimum, the size of the council? Cheers Sébastien ---- solutions of the puzzle authors below, without the justification (which I can make available in French if needed) ----- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ---- First problem: 1358 Second problem: 14