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Atraxani
 Expert Boarder
Posts: 83 
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Enigma 1298 - Odd change New Scientist magazine, 17 July 2004. by Susan Denham.
I have some coins in my purse whose total value is less than 1 pound stirling. 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?
Ciao,
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saintthomas
Expert Boarder
Posts: 89
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On 17 Sep 2004 01:28:55 -0700, Peter Chapman
Unless I am missing some trick the answer has to be 99p and most of the information given is not necessary to imply that conclusion. If the amount (say X) is equal to the number of possible totals, the possible totals must be 1 to X; if that is true after adding one pound, then the possible totals are then 1 to X+100; i.e. 99p is a possible total, so I must have had 99p before adding the pound. Lots of combinations work, for instance 1,2,2,2,2,10,10,10,10,50.
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klaretonor
 Senior Boarder
Posts: 74
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Mike Robson schrieb:
Well, so would 99 pennies, which must the combination with the most coins. Is yours the combination using the least? How many combinations are there?
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quest_marsman
Expert Boarder
Posts: 83
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Below is my list (not proven to be comprehensive) of the ways that the 99p can be made, while not permitting any duplication of any particular total. Have I missed any?
I thought that the problem would have been better if the solution was one particular combination of coins. Say, by specifying that the number of different coins and the total number of coins are both prime. The fact that Susan didn't make such a specification makes we wonder whether there is such a unique combination.
1p 2p 5p 10p 20p 50p #different coins #coins total
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Steve_Farmer_Jr
Senior Boarder
Posts: 70
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Peter Chapman schrieb:
Yes, there should be 21. I have amended your table below.
The combination of a penny and 2p and the combination of 4 pennies and 5p both fulfil your requirement (and the total is prime, too).
49 - - - - 1 2 50 19 - - - 4 - 2 23 9 - - - 2 1 3 12 4 - 1 - 2 1 4 8 1 4 - - 2 1 4 8
4 - 9 - - 1 3 14
The last two lines are duplicates, so there are 21 possible combinations in all.
Cheers
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Atraxani
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Posts: 83
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(duplicates removed)
Nice work Mendel. Did you do that by inspection or by computer proof?
So, what neat property could be used to specify one particular combination of coins? Preferably a combination that isn't too obvious?
Ciao,
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bhunders
Senior Boarder
Posts: 78
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Peter Chapman schrieb:
Inspection. You always need pennies, because you need to have the 1p total. There are 5 other denominations left, and you can just enumerate them binarily (2^5=32) and determine how many of each you need; and some combinations are impossible.
I have four times as many coins as I have denominations. (Leads you up the garden path nicely as no denomination has 4 coins. 
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