We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (other)

Prove that in any group of friends there will be two people who have the same number of friends.

Problems Methods Pigeonhole principle Pigeonhole principle (other)

In chess, ‘check’ is when the king is under threat of capture from another piece. What is the largest number of kings that it is possible to place on a standard \(8\times 8\) chess board so that no two check one another.

Problem #PRU-35543

Problems Algebra and arithmetic Word problems Combined labour (work) problems Set theory and logic Joke problem

11–12 2.0

One and a half diggers dig for a half hour and end up having dug half a pit. How many pits will two diggers dig in two hours?

Problem #PRU-35563

Problems Set theory and logic Theory of algotithms Theory of algorithms (other)

11–14 2.0

Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?

Problem #PRU-35571

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–13 2.0

A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.

Problem #PRU-35609

Problems Set theory and logic Joke problem

10–12 2.0

There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?

Problem #PRU-35615

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.

Problem #PRU-35627

Problems Methods Pigeonhole principle Pigeonhole principle (other) Set theory and logic Set theory Set theory (other) Algebra and arithmetic Fractions Simple fractions

12–15 3.0

A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?

Problem #PRU-35648

Problems Set theory and logic Mathematical logic Paradoxes

12–14 2.0

Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.

Problem #PRU-35662

Problems Set theory and logic Theory of algotithms Balance puzzles Methods Examples and counterexamples. Constructive proofs

12–13 2.0

What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.