I have the numbers 1..10 and want to arrange them in a billiard triangle such that they add up to the value on the right hand side thus. I can only use each number once.

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Is there a formula for working out the cominations? Any hints about proving it?

Alternatively: Knowing that ? must be 10 by Gauss' formula I have a list of combinations for each line:

line 2:


Line 3:


Line 4:

  • 1
    $\begingroup$ Your lists of combinations are not complete -- for example, $7+5+3=15$ and $7+6+2=15$ are missing $\endgroup$ Commented May 7, 2013 at 16:00

1 Answer 1


You've already got everything you need. You know the $10$ goes at the top; and you have disjoint lists of balls for the second and third rows, e.g. $9, 4$ and $8,6,1$. Then just put the remaining balls $2,3,5,7$ in the last row.

  • $\begingroup$ Thank you-. Is there a formula for this or is it really just writing out all the combinations and choosing them? Maybe a computer program to calculate the values? I can program matlab just need to know what I'm asking it to do. Also what happens if there are more constraints such as sums of the lines north to south? $\endgroup$
    – HCAI
    Commented May 7, 2013 at 16:22
  • $\begingroup$ @HCAI: That's a very broad question. Certainly if you have other constraints there might be some methods to fulfill them, but it's hard to say in that generality. And yes, you can certainly program a computer to find a solution; if you go through my answers you'll find I often write little programs to solve or investigate combinatorial and number-theoretical problems -- but again it's hard to say anything specific about how to do that -- one quite general method you should look at if you want to do that sort of thing is backtracking. $\endgroup$
    – joriki
    Commented May 7, 2013 at 16:31
  • $\begingroup$ Ok ignoring any further constraints, do you have any thoughts on a logical method of calculating the number combinations for each line? $\endgroup$
    – HCAI
    Commented May 7, 2013 at 17:46

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