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This is my first question on the Math SE, and it may be a bit simple. Apologies if my question reveals my misunderstanding of your field or the universe in general. Me Programmer.

I am working on a maths game to teach kids the use of multiplication, division and logic. Kids are given a sum of a set, and the values in a set. They need to show that they can work out the numbers in the set by dividing the sum into smaller parts (dividing by 1-10) and adding the product of that division to other products of that division. They can only use whole numbers.

For a really simple example, with a set of 30, 30 and 40 (and a sum of 100), can be solved by:

100/10          (split 100 into ten parts) 10+10+10        (add three lots of 10, etc.) 10+10+10         10+10+10+10      

Another example:

With a set of 18, 18, 12, and 6 and a sum of 54:

54/2    = 18 with 18 left over r18     = 18 r18/3*2 = 12 with 6 left over r6      = 6 

But I can't see a way to solve this with a Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 with the sum 376 

Or a set of prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23 with the sum 100 

Given the sum of a set of numbers, is there a relationship between the sum of the set and the numbers in the set (other than the obvious) that I can exploit to create problems for the kids to solve? Is there a way to test whether a set is solvable in this game, or a formula I should be using to test/generate the problems?

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    I'm afraid I don't understand what you want them to do. Can you either describe it more formally or give further examples? Is it a coincidence that all the summands in your example are the same ($10$), or is that required? If so, all the summands would have to be $1$ in the other two examples, since those numbers don't have any other common divisors.2012-08-02
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    Have you encountered the [coin problem](http://en.wikipedia.org/wiki/Coin_problem), by any chance?2012-08-02
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    @joriki Apologies I don't know how to describe it more formally; the 10 was just a coincidence. I've added another example to the question for you - let me know if you'd like more.2012-08-02
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    @J.M. the coin problem was really helpful, so were the McNugget numbers - I think that's the sort of area I need to look into.2012-08-02
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    @glenstorey: The example you added has only made things even more mysterious to me. Could you try to make more of an effort to give a more formal description of the task?2012-08-02

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