# Puzzles

## 18 December

The final round of game show starts with £1,000,000. You and your opponent take it in turn to take any value between £1 and £900.
At the end of the round, whoever takes the final pound gets to take the money they have collected home, while the other player leaves with nothing.

You get to take an amount first. How much money should you take to be certain that you will not go home with nothing?

## Turning squares

Source: Futility Closet

Each square on a chessboard contains an arrow point up, down, left or right. You start in the bottom left square. Every second you move one square in the direction shown by the arrow in your square. Just after you move, the arrow on the square you moved from rotates 90° clockwise. If an arrow would take you off the edge of the board, you stay in that square (the arrow will still rotate).

You win the game if you reach the top right square of the chessboard. Can I design a starting arrangement of arrows that will prevent you from winning?

## Placing plates

Two players take turns placing identical plates on a square table. The player who is first to be unable to place a plate loses. Which player wins?

## More doubling cribbage

Source: Inspired by Math Puzzle of the Week blog

Brendan and Adam are playing lots more games of high stakes cribbage: whoever
loses each game must double the other players money. For example, if Brendan has £3 and Adam has £4 then Brendan wins, they will have £6
and £1 respectively.

In each game, the player who has the least money wins.

Brendan and Adam notice that for some amounts of
starting money, the games end with one player having all the money; but for other amounts, the games continue forever.

For which
amounts of starting money will the games end with one player having all the money?

## Doubling cribbage

Source: Math Puzzle of the Week blog

Brendan and Adam are playing high stakes cribbage: whoever loses each game must double the other players money. For example, if Brendan has £3 and Adam has £4 then Brendan wins, they will have £6 and £1 respectively.

Adam wins the first game then loses the second game. They then notice that they each have £180. How much did each player start with?

## Twenty-one

Scott and Virgil are playing a game. In the game the first player says 1, 2 or 3, then the next player can add 1, 2 or 3 to the number and so on. The player who is forced to say 21 or above loses. The first game went like so:

Scott: 3

Virgil: 4

Scott: 5

Virgil: 6

Scott: 9

Virgil: 12

Scott: 15

Virgil 17

Scott: 20

Virgil: 21

Virgil loses.

To give him a better chance of winning, Scott lets Virgil choose whether to go first or second in the next game. What should Virgil do?