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Problems Discrete Mathematics Combinatorics Painting problems

A strange wonderland creature is called a painting chameleon. If the queen puts the painting chameleon on a chess-like board then he moves one square at a time along the board either horizontally or vertically. When he moves, he either changes his colour to the colour of the square he moves to, or he paints the square he moves to into his own colour. The queen puts a white painting chameleon on an all-black board $8 \times 8$ and orders the chameleon to paint the board into a chessboard. Can he succeed?

Problem #PRU-102844

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The cells of a $15 \times 15$ square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-109808

Problems Discrete Mathematics Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

Problem #PRU-110083

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

Can the cells of a $5 \times 5$ board be painted in 4 colours so that the cells located at the intersection of any two rows and any two columns are painted in at least three colours?

Problem #PRU-21979

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle

12–13 3.0

a) What is the maximum number of squares on an $8 \times 8$ grid that can be shaded in with a black pen such that each ‘L’ shaped group of 3 squares has at least one unshaded square.

b) What is the maximum number of squares on an $8 \times 8$ grid that can be shaded in with a black pen, such that each ‘L’ shaped group of 3 squares has at least one shaded square.

Problem #PRU-30815

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

12–13 3.0

Each of the edges of a complete graph consisting of 6 vertices is coloured in one of two colours. Prove that there are three vertices, such that all the edges connecting them are the same colour.

Problem #PRU-32068

Problems Discrete Mathematics 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-35238

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle Discrete Mathematics 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.

Problem #PRU-58080

Problems Discrete Mathematics Combinatorics Geometry on grid paper Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–13 3.0

A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.