Problems

Ten little circles are drawn on a squared board $4\times 4$.

Cut the board into identical parts in such a way that each part contains 1, 2, 3, and 4 drawn circles correspondingly.

Philip and Denis cut a watermelon into four parts. When they finished eating watermelon (they ate the whole thing), they discovered that there were five watermelon rinds left. How is it possible, if no rind was cut after the initial cutting?

Cut a square into a heptagon (7 sides) and an octagon (8 sides) in such a way, that for every side of an octagon there exists an equal side belonging to the heptagon.

Sometimes life can make us do the craziest of things. In this problem you just need to find out how one can cut an $8\times 8$ chessboard into 20 pieces each having the same perimeter and consisting of a whole number of cells.

Cut a rectangle into two identical pentagons.

(a) Cut the rectangle into two identical quadrilaterals.

(b) Cut the rectangle into two identical hexagons.

(c) Cut the rectangle into two identical heptagons.

a) You have a $10\times 20$ chocolate bar and 19 friends. Since you are good at maths they ask you to split this bar into 19 pieces (always breaking along the lines between squares). All the pieces have to be of a rectangular shape. Your friends don’t really care how much they will get, they just want to be special, so you need to split the bar in such way that no two pieces are the same.

(b) The friends are quite impressed by your problem solving skills. But one of them is not that happy with the fact you didn’t get a single piece of the chocolate bar. He thinks you might feel that you are too special, therefore he convinces the others that you should get another $10\times 20$ chocolate bar and now split it into 20 different pieces, all of rectangular shapes (and still you need to break along the lines between squares). Can you do it now?

Can one cut a square into (a) one 30-gon and five pentagons? (b) one 33-gon and three 10-gons?

Jennifer draws a hexagon, and a line passing through two of its vertices. It turns out one of the figures in which the original hexagon is divided is a heptagon. Show an example of a hexagon and a line for which it is true.

Can Jennifer draw an octagon and a line passing through two of its vertices in such a way that this line cuts a 10-gon from it?