Lightening the Load

pile of coins and notes

You may wish to look at the related problem Heavy Pockets first.

Imagine a world where we only had 1p coins. It would get really inefficient as soon as we needed to pay large amounts...

If we introduce a second coin, it might make life a bit easier. 

If we need to be able to pay any amount up to £2, without needing change, what could we choose as our second coin?

What's the best choice, if we want to minimise the total number of coins we have to carry?

What about if we needed to be able to pay any amount up to £3?


Is there a general rule? 

Which two denominations would it be best to choose if we needed to be able to pay any amount from 1p to Np?


What if we could choose 3 denominations of coin? Or 4? Or...