A wizard kidnaps 101 people
Source: Argentina TST 2011, Problem 2
August 29, 2014
combinatorics proposedcombinatoricswinning positions
Problem Statement
A wizard kidnaps members from party , members from party , members from party , and members from party , keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with or more people involved).a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.