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If there are

ncombinations and we add a pound coin there are nowncombinations (without the pound coin),ncombinations (with the pound coin) and the pound coin by itself, so 2n+ 1 combinations.And if the initial amount was

ppence, with the addition of the pound coin it is (100 +p) pence.But these are equal, so:

So it seems that if the problem has an answer the answer is 99p. We just need to find a combination of denominations of coins that sum to 99p that has has 99 different amounts that can be made in only one way (obviously the combinations will be all the values between 1p and 99p). One simple solution to this is 99× 1p.

This Python 3 program looks for all combinations of coins that sum to 99p in total and can make 99 different combinations. It runs in 284ms.

Solution:There is 99p in the purse.The program finds 18 different combinations of coins that satisfy the conditions of the problem.

The largest number of coins is 99× 1p (99 coins).

The smallest number of coins is 10, and there are 4 different ways of achieving this, e.g.: 1× 50p, 4× 10p, 1× 5p, 4× 1p.