# Puzzles

## 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?

## The taxman

Source: New York Times

In a very strange country, the tax system works as follows.

£1, £2, £3 up to £12 are available.

You pick an amount. You keep this amount, but the taxman takes any factors of it. You cannot pick any amount without a factor.

This continues until you can take no more money. The taxman gets any remaining money.

For example, you might play as follows:

- Take £12. Taxman gets £1, £2, £3, £4, £6.
- Take £10. Taxman gets £5.
- You cannot take anything. Taxman gets £7, £8, £9, £11.

In this example, you end with £22 and the taxman ends with
£56.

Is it possible to get more money than the taxman? What is the most you can get?

## No change

"Give me change for a dollar, please," said the customer.

"I'm sorry," said the cashier, "but I can't do it with the coins I have. In fact, I can't change a half dollar, quarter, dime or nickel."

"Do you have any coins at all?" asked the customer.

"Oh yes," said the cashier, "I have $1.15 in coins."

Which coins are in the cash register?

(The available coins are 50¢, 25¢, 10¢ 5¢ and 1¢.)

## Exact change

Source: @AlexDBolton on Twitter

In the UK, the coins less than £1 are 1p, 2p, 5p, 10p, 20p and 50p. How many coins would I need to carry in my pocket so that I could make any value from 1p to 99p?

In the US, the coins less than $1 are 1¢, 5¢, 10¢, 25¢. How many coins would I need to carry in my pocket so that I could make any value from 1¢ to 99¢?

## Pocket money

When Dad gave out the pocket money, Amy received twice as much as her first brother, three times as much as the second, four times as much as the third and five times as much as the last brother. Peter complained that he had received 30p less than Tom.

Use this information to find all the possible amounts of money that Amy could have received.

## Ninety nine

Source: UKMT Senior Maths Challenge 2013

In a 'ninety nine' shop, all items cost a number of pounds and 99 pence. Susanna spent £65.76. How many items did she buy?