We solve problems

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

Is it possible to bake a cake that can be divided by one straight cut into 4 pieces?

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties

Is it possible to cut a square into four parts so that each part touches each of the other three (ie has common parts of a border)?

Problem #PRU-88099

Problems Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

10–12 2.0

Twenty-eight dominoes can be laid out in various ways in the form of a rectangle of \(8 \times 7\) cells. In Fig. 1–4 four variants of the arrangement of the figures in the rectangles are shown. Can you arrange the dominoes in the same arrangements as each of these options?

Problem #PRU-88100

Problems Set theory and logic Mathematical logic Puzzles Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

10–13 2.0

An entire set of dominoes, except for 0-0, was laid out as shown in the figure. Different letters correspond to different numbers, the same – the same. The sum of the points in each line is 24. Try to restore the numbers.

Problem #PRU-88193

Problems Combinatorics

10–14 2.0

In an ordinary set of dominoes, there are 28 tiles. How many tiles would a set of dominoes contain if the values indicated on the tiles did not range from 0 to 6, but from 0 to 12?

Problem #PRU-88235

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Combinatorics Dissections, partitions, covers and tilings Various dissection problems

10–13 3.0

A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon (see the drawing). Is it possible to reconstruct the original square using just this information?

Problem #PRU-97823

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Methods Examples and counterexamples. Constructive proofs Combinatorics Painting problems

14–16 4.0

Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?

Problem #PRU-98016

Problems Combinatorics Integer lattices Integer lattices (other) Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

10–15 4.0

We are given 101 rectangles with integer-length sides that do not exceed 100.

Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).

Problem #PRU-102796

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

Problem #PRU-102844

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.