Maths circle: More Invariants and Tilings – We Solve Problems

Maths circle: More Invariants and Tilings

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.

Problems:

1.

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

Can you cover a $10 \times 10$ board using only $T$-shaped tetrominos?

2.

Invariants , Invariants and semi-invariants

A broken calculator can only do several operations: multiply
by 2, divide by 2, multiply by 3, divide by 3, multiply by 5, and divide by 5.
Using this calculator any number of times, could you start with the number
12 and end up with 49?

3.

Covers , Dissections, partitions, covers and tilings

Can you cover a $10 \times 10$ square with $1 \times 4$ rectangles?

4.

Covers , Dissections, partitions, covers and tilings

Two opposite corners were removed from an $8 \times 8$ chessboard. Can you cover this chessboard with $1 \times 2$ rectangular blocks?

5.

Invariants , Invariants and semi-invariants

The numbers 1 through 12 are written on a board. You can erase any two of these numbers (call them $a$ and $b$) and replace them with the number $a+b-1$. After 11 such operations, there will be just one number left. What could this number be?

6.

Invariants , Invariants and semi-invariants

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?

7.

Covers , Dissections, partitions, covers and tilings

One small square of a $10 \times 10$ square was removed. Can you cover the rest of it with 3-square $L$-shaped blocks?

8.

Invariants , Invariants and semi-invariants

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?

9.

Colouring , Colouring (other) , Dissections, partitions, covers and tilings

A $7 \times 7$ square was tiled using $1 \times 3$ rectangular blocks. One of the squares has not been covered. Which one can it be?

10.

Invariants , Invariants and semi-invariants

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?

11.

Covers , Dissections, partitions, covers and tilings

Can you cover a $13 \times 13$ square using two types of blocks: $2 \times 2 $ squares and $3 \times 3$ squares?

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