Problem

Source: 43rd International Tournament of Towns, Senior A-Level P1, Fall 2021

Tags: combinatorics, cards, Tournament of Towns



The wizards $A, B, C, D$ know that the integers $1, 2, \ldots, 12$ are written on 12 cards, one integer on each card, and that each wizard will get three cards and will see only his own cards. Having received the cards, the wizards made several statements in the following order. “One of my cards contains the number 8”. “All my numbers are prime”. “All my numbers are composite and they all have a common prime divisor”. “Now I know all the cards of each wizard”. What were the cards of $A{}$ if everyone was right? Mikhail Evdokimov