Write out the numbers 1-15 such that any two numbers in the series add up to a square number
Inspired by https://www.youtube.com/watch?v=G1m7goLCJDY
There are 6 possible squares that can be made by adding two numbers under 16:
- 1
- 2
- 4
- 9
- 16
- 25
Counting down from 15, here are all the possible squares that can be made with each number.
| 1 | 2 | 4 | 9 | 16 | 25 |
+-----+-----+-----+-----+------+-------+
| | | | | 15 1 | 15 10 |
| | | | | 14 2 | 14 11 |
| | | | | 13 3 | 13 12 |
| | | | | 12 4 | 12 13 |
| | | | | 11 5 | 11 14 |
| | | | | 10 6 | 10 15 |
| | | | 9 | 9 7 | |
| | | | 8 1 | | |
| | | | 7 2 | 7 9 | |
| | | | 6 3 | 6 10 | |
| | | | 5 4 | 5 11 | |
| | | 4 | 4 5 | 4 12 | |
| | | 3 1 | 3 6 | 3 13 | |
| | 2 | | 2 7 | 2 14 | |
| 1 | | 1 3 | 1 8 | 1 15 | |
+-----+-----+-----+-----+------+-------+
Most numbers can only make two different squares. Thus they have an obvious partner on each side (which side doesn't matter). Counting down from 15, start building chains of numbers between other numbers. When two chains end in the same number, connect them. Eventually you end with a single long chain.
By the time you reach the small numbers that have more than two possible partners, you've already already completed the chain.
Thus the answer is:
9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8