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