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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 \(8\times8\) and orders the chameleon to paint the board into a chessboard. Can he succeed?

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.

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.

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?

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.

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.

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?

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.

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.

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.