Maths circle: Parity and Invariants – We Solve Problems

Maths circle: Parity and Invariants

Today we will solve some problems involving counting the same quantity twice (or more times), in different ways, what will let us learn about the components making it up.

Problems:

1.

Invariants , Invariants and semi-invariants

Anna has $20$ novels and $25$ comic books on her shelf. She doesn’t really keep her room very tidy and so she also has a lot of novels and comic books in various places around her room. Each time she reaches for the shelf, she takes two books and puts one back. If she takes two novels or two comic books, she puts a novel back on the shelf. If she takes a novel and a comic book, she places another comic book on the shelf. That way, her shelf sistematically empties. Show that eventually there will be a lone comic book standing on her shelf and all her other books scattered across her room.

2.

Invariants , Invariants and semi-invariants

A knight in chess moves in an “L” pattern – two squares in one direction and one square in a perpendicular direction. Starting with a knight in the bottom right corner of a regular $8 \times 8$ chessboard, can you move it some number of times according to the rules in such a way that it visits every square on the chessboard exactly once and ends up in the top left corner?

3.

Invariants , Invariants and semi-invariants

Nine lightbulbs are arranged in a $3 \times 3$ square. Some are on, some are off. You are allowed to change the state of all the bulbs in a column or in a row. That means all the bulbs in that row or column that were off light up and the ones that were on go dark. Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?

 

4.

Invariants , Invariants and semi-invariants

Anna’s sister, Claire, has $10$ novels, $11$ textbooks and $12$ comic books on her shelf. She also doesn’t like to keep all her books there. Each time she takes two books of different type from the shelf and puts a book of the third type back on. So for example, she might take a novel and a comic book and put a textbook back. Show that eventually there is only a single textbook, and exactly a textbook, left on her shelf.

5.

Invariants , Invariants and semi-invariants

All the squares of a $9 \times 9$ chessboard were coloured black and white in a traditional way, such that the corner squares are all white. In each move, you can choose two neighbouring squares and change both of their colours – black to white and white to black. Can you reach a chessboard that is all black in this way?

6.

Invariants , Invariants and semi-invariants

a) A rook in chess can move any number of squares in the same row or column. Let’s invent a new figure, a “little rook” that can only move one square in each of these directions. If we start with a “little rook” in the bottom right corner of an $8 \times 8$ chessboard, can we make it to the opposite corner while visiting each square exactly once? $\\$
b) A king in chess moves like the “little rook”, but he can also move one square along a diagonal. Can we do the same task with a king?

7.

Invariants , Invariants and semi-invariants

There are $15$ lightbulbs in a row, they are all switched off. We can pick any three of them and change their state. Can we repeat this operation an \textbf{even} number of times such that at the end all the lightbulbs are on?

8.

Invariants , Invariants and semi-invariants

Are there such integer numbers $a,b,c,d,e,f$ that the numbers
$$ a -b, b-c,c-d,d-e,e-f,f-a$$
are consecutive integers? (note: they obviously can be negative)

9.

Invariants , Invariants and semi-invariants

There are numbers $1,2,3,4,5,6,7,8,9$ and $10$ written on a board. Each time you make a “move” you can erase three of the remaining numbers, $a,b$ and $c$, and replace them with the numbers $2a+b,2b+c$ and $2c+b$. The goal is to make all the $10$ numbers left on the board equal. Is it possible?

10.

Invariants , Invariants and semi-invariants

On a certain island there are 17 grey, 15 brown and 13 crimson chameleons. If two chameleons of different colours meet, both of them change to the third colour. No other colour changes are allowed. Is it possible that after a few such colour transitions all the chameleons have the same colour?

11.

Invariants , Invariants and semi-invariants

Sixteen lightbulbs are arranged in a $4 \times 4$ grid. Some are on, some are off. You are allowed to change the state of all the bulbs in a column, in a row, or along any diagonal (note: there are 14 diagonals in total!). Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?

 

12.

Invariants , Invariants and semi-invariants

There are numbers from $1$ to $2018$ written on a board. In one go, we can erase two numbers and replace them with an absolute value of their difference. Can we achieve a sequence consisting only of several numbers $0$ after some number of moves?

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