Suggestion: Find all possible neighbors of the number 16. (a) Show that the numbers from 1 to 16 can be written in a line, such that the sum of any two adjacent numbers is a perfect square.
(b) Show that the numbers from 1 to 16 cannot be written around a circle, such that the sum of any two adjacent numbers is a perfect square.
The key observation that helps solve (a) and resolves (b) is to look for the possible neighbors for the number 16.
A neighbor of 16 is a number that, when added to 16, results in a perfect square. One candidate is the number 9, since .
There are no others, because the next perfect square after 25 is 36, and the largest sum we can obtain from two numbers between 1 and 16 is .
(a) Since 16 has only one possible neighbor, it must be at an end. Starting with 16, we obtain the solution below.
(b) For it to be possible to place all the numbers from 1 to 16 around a circle, every number would need to have two neighbors. But the only possible neighbor for 16 is 9, making it impossible to construct a circular arrangement.