We solve problems

Problems Combinatorics Geometry on grid paper

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 Combinatorics Geometry on grid paper

12–13 2.0

How many squares are shown in the picture?

Problem #PRU-104058

Problems Combinatorics Geometry on grid paper

12–13 2.0

Cut the shape (see the figure) into two identical pieces (coinciding when placed on top of one another).

Problem #PRU-104089

Problems Combinatorics Geometry on grid paper Set theory and logic Algorithm Theory Game Theory Winning and loosing positions

11–14 3.0

There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

Problem #PRU-109808

Problems Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

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?

Problem #PRU-115384

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A game of ’Battleships’ has a fleet consisting of one \(1\times 4\) square, two \(1\times 3\) squares, three \(1\times 2\) squares, and four \(1\times 1\) squares. It is easy to distribute the fleet of ships on a \(10\times 10\) board, see the example below. What is the smallest square board on which this fleet can be placed? Note that by the rules of the game, no two ships can be placed on horizontally, vertically, or diagonally adjacent squares.

Problem #PRU-32075

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (area and volume) Dissections, partitions, covers and tilings Various dissection problems

12–14 4.0

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?

Problem #PRU-58080

Problems Combinatorics Geometry on grid paper Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–13 3.0

A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

Problem #PRU-60365

Problems Combinatorics Geometry on grid paper Painting problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.