Two people play a game like this. They take turns putting the same coins on the round table. You can't put coins on top of each other. The loser is the one who has nowhere to put the next coin. Which player is guaranteed to win: the beginner or his opponent? How should he play?

Leaflet #money

There are coins on the table. Fifteen of them are eagle up and the rest are eagle down. Blindfolded, you have to put these coins into two piles so that the number of coins lying eagle up in these piles is the same. The number of coins in the piles can be different (the pile can consist of any number of coins, including one or even less), the coins can be turned over, but it is impossible to determine by touch how a coin lies.

Leaflet #money

Cut the square into a) 4; b) 9; c) 17 squares.

Leaflet #cut

Four dwarfs inherited from their uncle an orchard enclosed by 16 matchsticks with 12 fruit trees. The location of the trees is shown in the drawing. Divide the garden using 12 matches into four equal parts containing an equal number of trees each. The matches are only allowed to be placed on the dotted lines. (Equal parts must have the same shape, size, and same arrangement of trees.)

Leaflet #cut

The watermelon was cut into 4 parts and eaten. It turned out five crusts. How could this be?

Leaflet #cut

Cut the figure shown in the figure into four equal parts.

Leaflet #cut

The square napkin was folded in half, the resulting rectangle was folded in half again. The resulting square was cut with scissors in a straight line. Could the napkin fall apart
a) into 2 parts?
b) into 3 parts?
c) into 4 parts?
d) into 5 parts?

Leaflet #cut

Cut the figure shown in the figure into four identical parts so that they can be folded into a 6×6 square with a chess coloring.

Leaflet #cut

1) Cut a rectangle 4×9 in two parts, from which you can add a square 6×6.

2) Cut a rectangle 9×16 in two parts, which can be folded into a square.

Leaflet #cut

Cut each of the following figures into two equal parts.

Leaflet #cut

Eight rabbits were put in seven cages. Prove that there is a cage in which there are at least two rabbits.

Leaflet #pigeonhole

More about the pigeonhole principle

A team of four received 10 candies for winning the math regatta. The children divided the candies among themselves without breaking them. Determine whether the following statements are true:
a) "someone got at least two sweets";
b) "someone got at least three sweets";
c) "two people got at least two sweets";
d) "everyone got at least one sweets".

Leaflet #pigeonhole

1) There is a closet with 24 black and 24 blue socks in a dark room. What minimum number of socks must be taken from the closet so that at least one pair of socks of the same color can be knowingly made up?

2) What is the minimum number of socks needed to knowingly make up at least one pair of black socks?

3) How will the solution of the problem change if there are 12 pairs of black and 12 pairs of blue socks in the box and you need to make up a pair of one color (as in 1) and a pair of black (as in 2)? ( Shoes, unlike socks, are left and right.)

Leaflet #pigeonhole

🥳 August 24: Birthday of Bartholomaeus Pitiscus, author of the term "trigonometry"

#celebration

There are a million spruces growing in the forest. It is known that each of them has no more than 600,000 needles. Prove that there are two spruces with the same number of needles.

Leaflet #pigeonhole

The school has 30 classes and 1000 students. Prove that there is a class with at least 34 students.

Leaflet #pigeonhole

In a square carpet with a side of 4 meters, the moth ate 15 holes. Prove that it is possible to cut a rug from this carpet with a side of 1 meter, in which there will be no holes.

Leaflet #pigeonhole

In the final of the school basketball championship, the team scored 9 goals. Prove that there are 2 players of this team who scored equally goals. (There are 5 players on the basketball team.)

Leaflet #pigeonhole

Is it true that there are at least two people in your room who have the same number of friends in that room?

Leaflet #pigeonhole

There are two buckets: one with a capacity of 4 liters and the other with a capacity of 9 liters. Is it possible to use them alone to draw exactly 6 liters of water from the river?

Leaflet #transfusion

1) Is it possible to draw 4 liters of water from the tap with two 3-liter and 5-liter jars?

2) The same question if there are only 6-liter and 9-liter jars?

Leaflet #transfusion