Question

How must one place the integers from 1 to 15 in each of the spaces below in such a way that no number is repeated and the sum of the numbers in ANY TWO consecutive spaces is a perfect square?

Note: the sum can repeat,like u can have the sum as 9 as many times as u want.

Answer 1

perfect squares are
1,4,9,16,25,36
obviously cannot add to 1 or 36
4,9,16,25 are the sums to add up 2

ok so
8,1,15,10,6,3,13,12,4,5,11,14,2,7,9
after lots of trial and error

Answer 2

The perfect squares 1, 4, 9, 16, and 25 are possible in this problem since the biggest numbers which are 14 and 15 add up to 29 only.

A possible sequence would be,
9 - 7 - 2 - 14 - 11 - 5 - 4 - 12 - 13 - 3 - 6 - 10 - 15 - 1 - 8