We solve problems

Problems Set theory and logic Mathematical logic Puzzles

In the line of numbers and signs \({}* 1 * 2 * 4 * 8 * 16 * 32 * 64 = 27\) position the signs “\(+\)” or “\(-\)” instead of the signs “\(*\)”, so that the equality becomes true.

Problem #PRU-103850

Problems Set theory and logic Mathematical logic Puzzles

12–13 2.0

The code of lock is a two-digit number. Ben forgot the code, but he remembers that the sum of the digits of this number, combined with their product, is equal to the number itself. Write all possible code options so that Ben could open the lock quickly.

Problem #PRU-103858

Problems Algebra and arithmetic Number theory. Divisibility The fundamental theorm of arithmetic. Prime factorisation.

12–13 2.0

Solve the rebus: \(AX \times UX = 2001\).

Problem #PRU-103869

Problems Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Solve the puzzle: \(BAO \times BA \times B = 2002\).

Problem #PRU-103955

Problems Set theory and logic Theory of algotithms Game theory Winning and loosing positions

12–13 2.0

Ben and Joe play chess. In addition to a chessboard, they have one rook, which they put in the lower right corner, and they move it in turns. It can only be moved upwards or to the left (for any number of cells). The player who can not make a move, loses. Joe goes first. Who will win with the correct method?

Problem #PRU-103960

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Prove that if 21 people collected 200 nuts between them, there are two people in the group who collected the same number of nuts.

Problem #PRU-104001

Problems Combinatorics Dissections, partitions, covers and tilings Various dissection problems

12–13 2.0

Kai has a piece of ice in the shape of a “corner” (see the figure). The Snow Queen demanded that Kai cut it into four equal parts. How can he do this?

Problem #PRU-104009

Problems Set theory and logic Mathematical logic Mathematical logic (other)

12–13 2.0

The king made a test for the future groom of his daughter. He put the princess in one of three rooms, a tiger in the other, and left the last room empty. It is known that the sign on the door where the princess is sitting is true, where the tiger is – it is false, and nothing is known about the sign on the third room. The tablets are as follows:

1 – room 3 is empty

2 – the tiger is in room 1

3 – this room is empty

Can the prince correctly guess the room with the princess?

Problem #PRU-104021

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Algebra and arithmetic Number theory. Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

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.

Problem #PRU-104029

Problems Combinatorics Geometry on grid paper

12–13 2.0

a) In the construction in the figure, move two matches so that there are five identical squares created. b) From the new figure, remove 3 matches so that only 3 squares remain.