A difference pyramid

Here is a “simple” problem my 7-year-old son was given for homework.

The task is to put the numbers 1 to 10 into the pyramid so that each number appears once and is the difference of the two numbers in the blocks below it.

I have started to fill in the pyramid. You can see that 1 is the difference of 3 and 4. **This is for explanation only and you should start from an empty pyramid.***{format changed to bold after publication -jb}*

If you cannot find a solution, can you prove that there is no solution? Can you derive any constraints?

This is tough one. I solved it after about 2 hours. Good luck

James

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Duncan Corps

said:Hmm, a nice puzzle, the first constraint which occurs to me is that 10 must be on the lowest level since it can’t be the difference between any of the other available numbers.

And since 1, 3 and 4 are already used, that means that the other number on the lowest level can’t be 6, 7 or 9.

That means that the fourth number on the lowest level must be 2, 5 or 8. Can’t be 5 because the difference between 10 and 5 is 5, can’t use it twice 🙂 . So, 2 or 8 then.

Dr James Bayley

said:Sorry Duncan, The numbers 1,3,4 are only given as an example. They may or may/not form part of the actual solution.

Duncan Corps

said:D’oh! Didn’t spot that. Still puzzling over it when I have a few moments. I feel that 5 is important for some reason . Probably because of its relationship to 10 and itself…

Dr James Bayley

said:That is an interesting point. We now know that 10 is on the first row and that 5 cannot be next to it on the first row nor above it on the second row.

Dr James Bayley

said:In response to your comment I have edited the original post to improve clarity.

Phil

said:Hi James,

I found an answer using the constraint about 10 being on the bottom row, a couple of ‘ifs’ regarding the 8 and 9 and a spreadsheet.

It only took about 10 mins on the spreadsheet. Is that cheating?

my solution is:

4

1 5

6 7 2

9 3 10 8

Phil

Dr James Bayley

said:Well done. I don’t think using a spreadsheet is cheating – unless like me you use the Excel Analysis pack to find a solution by brute force! However it is more satisfying if we can find a solution by analysis or at least use analysis or an heuristic to reduce the number of permutations that we need to consider from Permute(3,9)*2 (given that 10 is on the first row and has two possible positions) to a much smaller set.