We had an amazing response to this problem, with everyone who submitted a solution getting the 17 numbers in the right order, that is:
\[16\quad 9\quad 7\quad 2\quad 14\quad 11\quad 5\quad 4\quad 12\quad 13\quad 3\quad 6\quad 10\quad 15\quad 1\quad 8\quad 17\]
We also received fantastic explanations, Ashleigh from the Sirius Academy used a systematic approach:
I started by listing all the square numbers up to 36, I then worked out all the addition pairs using a systematic approach to make sure I didnt miss any. I used this to make a separate list of all the pairs that made a square number when added together. I used a trial and improvement method to work out the order of the line.
Sam from Acland writes:
I worked out that 17 + 16 made 33, this means the highest square number I could make was 25. This means 17 and 16 only had one way to make a square number, so they would be at either end. After 17 it must be 8 and, after 8, 1. At this point there are two options, 3 or 15. But if 15 didn't go with 1 then it would have to go at the end which would have been impossible to complete. After this the problem became simple as from this point it becomes linear with only one option available after every point.
Christian and Pippa worked along similar lines:
We have started to solve this problem by finding out what numbers go with only one other number. Those are 16 and 17 and they had to go at the end. Then we found all the numbers that go with only 2 other numbers. Once we found that, we matched the remaining numbers. This can be the only solution because if we swapped one number, it would make a cut in the line and it wouldn't match. For example: 3 can go with more than 2 numbers. It can go with 13, 6 and 1 but if we put 1 next to 3, it wouldn`t work because 15 can only go with 1 and 10.
Robin from Oxford noticed this too:
I started by listing all pairs of numbers that added together to make a square number. Then I took the numbers that only added up with one other number to make a square number (16 and 17)and put them at the ends of the sequence. The two numbers that added up with 16 and 17 to make a square number (9 and 8) went next to 16 and 17 and so on.
Emma, Joe and Lucy Attwood, James Nash, Nathan Wilson, Pavan Murali, Sara Jafar, Rachel Musgrave, Philip Knott, Tom Short, Matthew Clark, Alasdair Haines, Arjun Gill and Henry McEntyre and gave fantastic explanations, well done!
Rob from Thurston explains why there is only one solution,
17 can only be paired with 8 to make 25 so has to be at one end but it doesn't matter which end it is at as all the numbers add up the same if you just swap them round.
Mrs Dillon's year 7 class also noticed a pattern:
We have noticed that the square numbers produced form a pattern: 25, 16, 9, 16, 25, 16, 9 etc.