*Spoiler for @standupmaths puzzle 24 Sep 13 *(Follow the link to see the puzzle.)

This is a quickie. Most of the numbers from 1 to 17 have only two other numbers in that range they can add to to get a square. For instance the numbers from 9 to 15 can add to the numbers from 7 to 1 (to get 16) or the numbers from 16 to 10 (to get 25) and nothing else. The only exceptions are 1 and 3, which can add to three other numbers, and 16 and 17, which can add to only one.

So right away we know the sequence has to start with 16 and end with 17 (or vice versa). The number after 16 has to be 9, and now that we’ve used 16 there’s only one number 9 can add to, which is 7. And so on until we hit 13 followed by 3, when we have two possibilities: 1 and 6. But working backward from 17 we find 17 is preceded by 8 is preceded by 1, with 6, 10 and 15 unused. So 3 must be followed by 6, then 10, then 15, then 1 and on from there to 17:

16 – 9 – 7 – 2 – 14 – 11 – 5 – 4 – 12 – 13 – 3 – 6 – 10 – 15 – 1 – 8 – 17

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