We solve problems

Problem #PRU-111328

Problems Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.

Problem #PRU-111682

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Proof by contradiction

12–14 3.0

100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.

Problem #PRU-111770

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

In 25 boxes there are spheres of different colours. It is known that for any \(k\) where \(1 \leq k \leq 25\) in any \(k\) of the boxes there are spheres of exactly \(k+1\) different colours. Prove that a sphere of one particular colour lies in every single box.

Problem #PRU-115384

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A game of ’Battleships’ has a fleet consisting of one \(1\times 4\) square, two \(1\times 3\) squares, three \(1\times 2\) squares, and four \(1\times 1\) squares. It is easy to distribute the fleet of ships on a \(10\times 10\) board, see the example below. What is the smallest square board on which this fleet can be placed? Note that by the rules of the game, no two ships can be placed on horizontally, vertically, or diagonally adjacent squares.

Problem #PRU-115449

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–16 4.0

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an \(8\times 8\) square grid, so that no more ’corners’ would fit?

Problem #PRU-115454

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

13–15 4.0

An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.

Problem #PRU-115467

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

12–15 4.0

10 children, including Billy, attended Billy’s birthday party. It turns out that any two children picked from those at the party share a grandfather. Prove that 7 of the children share a grandfather.

Problem #PRU-115470

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

A class has 25 pupils. It is known that for any two girls in the class, the number of male friends they have in the class is different. What is the maximum number of girls that it is possible for there to be in the class?

Problem #PRU-115489

Problems Set theory and logic Mathematical logic Mathematical logic (other) Methods Algebraic methods Proof by exhaustion

10–12 2.0

One day, Claudia, Sofia and Freia noticed that they brought the same toy cars to kindergarten. Claudia has a car with a trailer, a small car and a green car without a trailer. Sofia has a car without a trailer and a small green one with a trailer, and Freia has a big car and a small blue car with a trailer. What kind of car (in terms of colour, size and availability of a trailer) did all of the girls bring to the kindergarten? Explain the answer.

Problem #PRU-115492

Problems Methods Algebraic methods Proof by exhaustion Set theory and logic Mathematical logic Puzzles

11–13 2.0

The \(KUB\) is a cube. Prove that the ball, \(CIR\), is not a cube. (\(KUB\) and \(CIR\) are three-digit numbers, where different letters denote different numbers).