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Found: 33

Problem #PRU-32068

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

11–13 3.0

A square area of size $100 \times 100$ is covered in tiles of size $1 \times 1$ in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

Problem #PRU-35151

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other)

14–16 3.0

Every point in a plane, which has whole-number co-ordinates, is plotted in one of $n$ colours. Prove that there will be a rectangle made out of 4 points of the same colour.

Problem #PRU-35238

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.