We solve problems

Problems Set theory and logic Order relations

Uncle Jack, the cat Whiskers, Spot and postman Pat are sitting on a bench. If Spot, sitting to the right of everyone, sits between Uncle Jack and the cat, then the cat will be at the extreme left. In what order do they sit?

Problem #PRU-103778

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

11–11 2.0

Cut a square into three pieces, from which you can construct a triangle with three acute angles and three different sides.

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-103781

Problems Fun Problems Word Problems

11–12 2.0

A pedestrian walked along six streets of one city, passing each street exactly twice, but could not get around them, having passed each one only once. Could this be?

Problem #PRU-103784

Problems Set theory and logic Mathematical logic Puzzles

11–11 2.0

When Harvey was asked to come up with a problem for the mathematical Olympiad in Sunny City, he wrote a rebus (see the drawing). Can it be solved? (Different letters must match different numbers).

Problem #PRU-103787

Problems Methods Algebraic methods Counting in two ways Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.

Problem #PRU-103800

Problems Set theory and logic Joke problem

10–12 2.0

In two purses lie two coins, and one purse has twice as many coins as the other. How can this be?

Problem #PRU-103802

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Divisibility of a number. General properties Set theory and logic Set theory Set cardinality. One-to-one correspondence

12–13 2.0

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?

Problem #PRU-103812

Problems Algebra and arithmetic Arithmetic operations. Number identities Set theory and logic Mathematical logic Puzzles

11–11 2.0

Alex laid out an example of an addition of numbers from cards with numbers on them and then swapped two cards. As you can see, the equality has been violated. Which cards did Alex rearrange?

Problem #PRU-103815

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

11–11 2.0

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.