We solve problems

Problem #PRU-103780

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other) Proof by exhaustion

11–13 3.0

In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be: 1) 15 classmates? 2) 16 classmates?

Problem #PRU-103882

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

11–12 2.0

Find the smallest four-digit number \(CEEM\) for which there exists a solution to the rebus \(MN + PORG = CEEM\). (The same letters correspond to the same numbers, different – different.)

Problem #PRU-103888

Problems Methods Algebraic methods Proof by exhaustion Geometry Solid geometry Visual geometry in space

11–12 2.0

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

Problem #PRU-110920

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebraic methods Proof by exhaustion

11–13 2.0

Peter has some coins in his pocket. If Peter pulls \(3\) coins from his pocket, without looking, there will always be a £1 coin among them. If Peter pulls \(4\) coins from his pocket, without looking, there will always be a £2 coin among them. Peter pulls \(5\) coins from his pocket. Identify these coins.

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).

Problem #PRU-30738

Problems Algebra Number theory. Divisibility Divisibility rules Divisibility tests for 2 and 4 Methods Algebraic methods Proof by exhaustion

11–12 2.0

How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,

a) if each number can occur only once?

b) if each number can occur several times?

Problem #PRU-64494

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

10–11 2.0

Specify any solution of the puzzle: \(2014 + YES =BEAR\).

Problem #PRU-78629

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebraic methods Proof by exhaustion

12–14 3.0

Is it possible to place the numbers \(1, 2,\dots 12\) around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

Problem #PRU-79505

Problems Methods Pigeonhole principle Algebraic methods Proof by exhaustion

12–14 3.0

In March 2015 a teacher ran 11 sessions of a maths club. Prove that if no sessions were run on Saturdays or Sundays there must have been three days in a row where a session of the club did not take place. The 1st March 2015 was a Sunday.