Find a solution of the equation \(x^2 + 2017 = y^2\) in integer numbers.
Alice marked several points on a line. Then she put more points – one point between each two adjacent points. Show that the total number of points on the line is always odd.
One sunny day Alice met the White Rabbit. The Rabbit told her that he owns a pocket watch which has 11 gears arranged in a chain loop. The rabbit asked Alice if it was possible for all the gears to rotate simultaneously. What is your opinion on this matter? Can all the gears rotate simultaneously?
After the Mad Tea-Party, the Hatter was so excited that he decided to cool down by going on a short walk across the chessboard. He started at position a1, then walked around in steps taking each step as if he was a knight, and eventually returned back to a1. Show that he made an even number of steps.
The Cheshire Cat wrote one of the numbers \(1, 2,\dots, 15\) into each box of a \(15\times15\) square table in such a way, that boxes which are symmetric to the main diagonal contain equal numbers. Every row and column consists of 15 different numbers. Show that no two numbers along the main diagonal are the same.
Is it possible to divide the numbers 1, 2, 3, ..., 100 into pairs of one odd and one even number, such that in every pair except one the even number is greater than the odd number
Petya and Misha play such a game. Petya takes in each hand a coin: one – 10 pence, and the other – 15. After that, the contents of the left hand are multiplied by 4, 10, 12 or 26, and the contents of the right hand – by 7, 13, 21 or 35. Then Petya adds the two results and tells Misha the result. Can Misha, knowing this result, determine which hand – the right or left – contains the 10 pence coin?
The parliament of a certain country has two houses with an equal number of members. In order to make a decision on an important issue all the members voted and there were no abstentions. When the chairman announced that the decision had been taken with a 23-vote advantage, the opposition leader declared that the results had been rigged. How did he know it?
a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?
b) The same question, but if 1 and \(-1\) are used instead.
Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?