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Good day to all of you,

I don't know if this is basic question or not, and I might have phrase the question wrong, Please pardon me.

The question was

The numbers {1,2,...,8} can be paired so that the sum of each pair is a square number:

1+8 =9, 2+7=9, 3+6=9, 4+5=9

a) Prove that you can also do this with the numbers {1,2,...,16}

b) Prove that you cannot do this with the number {1,2,...12}

Please help, and thank you before hand for any help or hint at all.

Update: I try to go around with different series and apparently, if you add of all the number in the series, and its' factor is a squared number then the series can be paired such that the sum of each paired is a squared number. (of course, the total number of numbers must be even)

If you take the series {1,2,3,...,32} the sum would be 528 which is divisible by 16. Extend that series until 48 and you can still paired each number such that the sum is a squared number.

For labour illustration, from the series of {1,2,..32}

I sub divide it for this one into {1,2,...,8},{9,10,...16},{17,18,...32}

for each add the first and the last, work your ways inward.

If you find any flaw in my finding, please help! Thanks

cong le
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    In b), you can't even pair the numbers, as there are $13$ of them, and $13$ is an odd number. Are you actually having trouble with a)? – Gerry Myerson Apr 03 '23 at 07:18
  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Apr 03 '23 at 07:18
  • What can 1 pair with? What can 2 pair with?... – paw88789 Apr 03 '23 at 09:27
  • So, cong, have you solved the $16$ case? – Gerry Myerson Apr 03 '23 at 09:59
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    @GerryMyerson: Yes as of the update thought of solution, if you add from 1 to 16 together, the total is 136 which is divisible by 4. If you paired them up, the paired would be 1 to 8 just like the series {1,2,...8} while 9 to 16 add first to last and slowly come to the middle. – cong le Apr 03 '23 at 10:24
  • I'm sorry @paw88789, but I don't get what you mean. If you are thinking of a series consist of just those number then it is not possible to pair them up at all. If taken from my hypothesis, then the smallest possible sum of all number in a series would be 16. – cong le Apr 03 '23 at 10:27
  • Note that this is closely related to https://math.stackexchange.com/questions/1676413/for-which-n-can-1-2-n-be-rearranged-so-that-the-sum-of-each-two-adja and to some questions linked from there. – Gerry Myerson Apr 03 '23 at 13:26
  • For b, note that the highest sum of two elements is $11 + 12 = 23$, and the lowest sum possible is $3$, so the only squares that can be a sum are ... – Paul Sinclair Apr 04 '23 at 03:27
  • According to https://oeis.org/A253472 there is such a pairing for the numbers $1,2,\dots,2n$ for $n=4,7,8,9$, and all $n\ge12$. – Gerry Myerson Apr 06 '23 at 06:19

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