Some forty years ago my wife Christine and I considered the problems in this article, which involves a fair amount of computation. Computing facilities were not good then, so we considered instead the problems in  in which we showed, without using computers, that there were 64703 ways to make up £1 using coins; this was before the introduction of the 20p and £1 coins, and the ½p coin was in circulation. If Christine were still with us, this would have been another piece of joint work. I therefore dedicate this article to her memory.
The design of a coinage system depends on considerations we give to various criteria: for example, the number of denominations for the coins, the maximum number of coins required to deliver any given amount in a range, or the required number of coins averaged over the range; see also §3.