Problem

Source: (2021 -) 2022 Dürer Math Competition Regional E+3 https://artofproblemsolving.com/community/c1621671_

Tags: combinatorics



a) A game master divides a group of $40$ players into four teams of ten. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.) b) On the next occasion, the game master writes the names of $7$ teammates and $2$ opposing players on each card (possibly in a mixed up order). Now he wants to write the names in such a way that the players together cannot determine the four teams. Is it possible for him to achieve this? c) Can he write the names in such a way that the players together cannot determine the four teams, if now each card contains the names of $6$ teammates and $2$ opposing players (possibly in a mixed up order)?