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Programming Enigma Puzzles

27 August 2014

Posted by on **From New Scientist #2456, 17th July 2004**

I have some coins in my purse whose total value is less than 1 pound sterling. I have tried to make various totals using one or more of these coins and I have discovered two interesting facts.

First, each possible total can only be achieved by one particular combination of denominations. Secondly, the number of different totals possible equals the total value of the coins in pence. If I added any number of 1 pound coins to my purse those two facts would still be true.

Current UK coins with a value less than 1 pound are 50p, 20p, 10p, 5p, 2p, and 1p.

How much money do I have in total in my purse?

**Note:** I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

[enigma1298]

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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.