Square Termini
The first sixteen positive integers are rearranged into a sequence with sixteen terms such that the sum of any two consecutive terms is a perfect square.
What is
The answer is 24.
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1 solution
All the integers in the sequence can form perfect squares when summed with one of at least two of the other (distinct) integers in the sequence except for 8 , which only forms a perfect square when summed with 1 , and 1 6 , which only forms a perfect square when summed with 9 . So if such a rearrangement exists, 1 6 and 8 must be situated at either end. So now we just need to establish that such a rearrangement exists, and the following, (or its reverse), suffices, (uniquely, in fact):
1 6 , 9 , 7 , 2 , 1 4 , 1 1 , 5 , 4 , 1 2 , 1 3 , 3 , 6 , 1 0 , 1 5 , 1 , 8
The desired answer is therefore 1 6 + 8 = 2 4 .