How to solve is probably more important than just solving, so here's how I did it. First write down the perfect squares up to 36: 4, 9, 16, 25, 36. These are the numbers (which we can use more than once) as the sum of our adjacent pairs. We know we have to use 17 somewhere and the perfect square nearest to it and bigger than it is 25. 36 is too far away. So we must start the sequence with 17 and 8. The next number must be 1, because we'd need another 8 to get 16 and we've already used 17 to get 25. Now we have a choice: we could use 3 or 15 to get 4 or 16. This is where we create a tree. One branch goes to 3 and the other to 15. We follow each branch to see where it leads. Take the 3-branch first: we need 13 to get to 16 and we need 12 to get 25, then 4, 5, 11, 14, 2, 7, 9, 16 and we can't go any further. OK, now look at the 15-branch. We need 10 to get 25. Then we need 6 to get 16 and 3 to get 9. But we already have a 3-branch, so we can join the 15-branch to it using 6 as the link. That's it! We've used up all the numbers from 1 to 17! Here they are: 17, 8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16.