We already did some problems using invariants, quantities that do not change as a result of some action and can be used to show that certain states cannot be achieved. This time, we will solve some more problems on invariants and learn about another type of invariants, that can tell us whether a board can be tiled with tiles of specific shapes.
If a magician puts 1 dove into his hat, he pulls out 2 rabbits and 2 flowers from it. If the magician puts 1 rabbit in, he pulls out 2 flowers and 2 doves. If he puts 1 flower in, he pulls out 1 rabbit and 3 doves. The magician starts with 1 rabbit. Could he end up with the same number of rabbits, doves, and flowers after performing his hat trick several times?
There are real numbers written on each field of a $m \times n$ chessboard. Some of them are negative, some are positive. In one move we can multiply all the numbers in one column or row by -1. Is that always possible to obtain a chessboard where sums of numbers in each row and column are nonnegative?
Tom found a large, old clock face and put 12 sweets on the number 12. Then he started to play a game with himself. In each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly 1 of the sweets on each number?