## Counting Coins

#### Problem

In the United Kingdom, money is made up of pounds (£) and pence (p). The coins in circulation are 1p, 2p, 5p, 10p, 20p, 50p, £1 (100p), and £2 (200p).

Surprisingly it is possible to have £2.39 worth of coins, made up of 100p + 50p + 4 20p + 9 1p, and be unable to make exactly £2.

What is the maximum amount of coinage that you could have in your pocket and not able to make exactly £2?

#### Solution

That maximum amount of money that you can have in your pocket, and be unable to make exactly £2, is £2.43 (243p). This can be achieved in three different ways:

- 100p + 50p + 4 20p + 5p + 4 2p (11 coins)
- 3 50p + 4 20p + 5p + 4 2p (12 coins)
- 50p + 9 20p + 5p + 4 2p (15 coins)

Using 1p, 2p, 5p, and 10p coins, what is the maximum amount of money you could have and be unable to make 20p?

What is the maximum amount of coins to achive this?

What about using 1p, 2p, 5p, 10p, and 20p coins, so that you cannot make 50p?

Investigate for different maximum amounts.