We solve problems

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

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 Discrete Mathematics 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 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 Discrete Mathematics 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.

Problem #PRU-104030

Problems Discrete Mathematics Combinatorics Geometry on grid paper

12–13 2.0

How many squares are shown in the picture?

Problem #PRU-104037

Problems Algebra and arithmetic Arithmetics. Mental maths Fun Problems Joke Problems

11–13 2.0

Gerard says: the day before yesterday I was 10 years old, and next year I will turn 13. Can this be?

Problem #PRU-104039

Problems Fun Problems Joke Problems

12–14 2.0

We call a natural number “amazing” if it has the form \(a^b + b^a\) (where \(a\) and \(b\) are natural numbers). For example, the number 57 is amazing, since \(57 = 2^5 + 5^2\). Is the number 2006 amazing?

Problem #PRU-104041

Problems Fun Problems Joke Problems

12–13 2.0

Homework. Cut a hole in an exercise book of a size so that you yourself can climb through it.

Problem #PRU-104045

Problems Fun Problems Word Problems

12–13 2.0

a) In how many ways can Dima paint five Christmas trees in silver, green and blue colours, if the amount of paint is unlimited, and he paints each tree in only one colour?

b) Dima has five baubles: a red, a green, a yellow, a blue and a gold one. In how many ways can he decorate five Christmas trees with them, if he needs to put exactly one bauble on each tree?

c) What about if he can hang several baubles on one Christmas tree (and all of the baubles have to be used)?

Problem #PRU-104046

Problems Discrete Mathematics Combinatorics

10–13 2.0

A toddler has \(25\) lego pieces in a box:

In how many ways are there to choose three pieces to play with?
In how many ways can he choose three pieces for the foundation, main walls and roof? Note that the order is important.