We solve problems

Problem #PRU-65194

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Methods Pigeonhole principle Pigeonhole principle (area and volume)

14–15 4.0

Every day, James bakes a square cake size $3 \times 3$ . Jack immediately cuts out for himself four square pieces of size $1 \times 1$ with sides parallel to the sides of the cake (not necessarily along the $3 \times 3$ grid lines). After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?

Problem #PRU-65718

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

In a $10 \times 10$ square, all of the cells of the upper left $5 \times 5$ square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)

Problem #PRU-87939

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Set theory and logic Set theory Unusual constructions

10–14 2.0

Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?

Problem #PRU-88084

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

10–12 2.0

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-102845

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

11–12 2.0

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).

Problem #PRU-110072

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?