Suppose the positive integer \(n\) is odd.

First Al writes the numbers \(1,2, \ldots, 2 n\) on the blackboard.

Then he picks any two numbers \(a, b\), erases them, and writes, instead, \(|a-b|\).

Prove that an odd number will remain at the end.

In the Parliament of Sikinia, each member has at most three enemies.

Prove that the house can be separated into two houses, so that each member has at most one enemy in his own house.

\(2n\) ambassadors are invited to a banquet.

Every ambassador has at most \(n-1\) enemies.

Prove that the ambassadors can be seated around a round table, so that nobody sits next to an enemy.

Assume that being enemies is mutual.

There are \(a\) white, \(b\) black, and \(c\) red chips on a table.

In one step, you may choose two chips of different colors and replace them by a chip of the third color.

If just one chip will remain at the end, its color will not depend on the evolution of the game.

When can this final state be reached?

Let's Play

Assume an 8 x 8 chessboard with the usual coloring.

You may repaint all square (a) of a row or column (b) of a 2 x 2 square.

The goal is to attain just one black square. Can you reach the goal?

Let's Play

Each of the numbers \(a_1, \ldots, a_n\) is 1 or -1, and we have

$$S=a_1 a_2 a_3 a_4+a_2 a_3 a_4 a_5+\cdots+a_n a_1 a_2 a_3=0$$

Prove that \(4 \mid n\).

The following game is played on an infinite chessboard. Initially, each cell of an \(n \times n\) square is occupied by a chip.

A move consists in a jump of a chip over a chip in a horizontal or vertical direction onto a free cell directly behind it.

The chip jumped over is removed. Find all values of \(n\), for which the game ends with one chip left over.