Solution to: Square Sequence
We first look with which numbers each number can form a pair of which the sum is a square:
| Number: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| Square pairs: | 1 + 3 1 + 8 1 + 15 | 2 + 7 2 + 14 | 3 + 1 3 + 6 3 + 13 | 4 + 5 4 + 12 | 5 + 4 5 + 11 | 6 + 3 6 + 10 | 7 + 2 7 + 9 | 8 + 1 | 9 + 7 9 + 16 | 10 + 6 10 + 15 | 11 + 5 11 + 14 | 12 + 4 12 + 13 | 13 + 3 13 + 12 | 14 + 2 14 + 11 | 15 + 1 15 + 10 | 16 + 9 |
As can be seen, there are two numbers that can form a pair in only one way: 8 and 16. Therefore, these two numbers need to be at the start or at the end of the sequence. If we start with the number 16, the sequence follows almost automatically: 16 9 7 2 14 11 5 4 12 13 3 6 10 15 1 8. Of course, the reverse sequence is also a possible solution.
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