Italian mathematical olympiad 2002, problem 6
Source:
February 13, 2012
graph theorycombinatorics proposedcombinatorics
Problem Statement
We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white.a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares?b) Is it possible to colour some squares in such a way that an odd number of them have exactly adjacent coloured squares and all the remaining coloured squares have exactly adjacent coloured squares?c) Is it possible to colour some squares in such a way that an odd number of them have exactly adjacent coloured squares and all the remaining coloured squares have exactly adjacent coloured squares?