MathNet 2002 — Part 01 Question 145

Let us give a chessboard of 100 rows and 100 columns, with all squares white.


a. Is it possible to color an odd number of boxes such that each colored box has an odd number of adjacent colored boxes?


b. Is it possible to color some squares such that an odd number of them have exactly 4 colored adjacent squares and all other colored squares have exactly 2 colored adjacent squares?


c. Is it possible to color some squares such that an odd number of them have exactly 2 colored adjacent squares and all other colored squares have exactly 4 colored adjacent squares?

Note: Two spaces are considered adjacent if they have a common side.