April 2026 Problem of the Month

Think of a positive number 𝑛.
Add up all the numbers from 1 to n (this is called a triangular number).
When does the total form a perfect square?

For example:
If n=1, the sum is 1 -- a perfect square.
If n=4, the sum is 10 -- not a perfect square
If n=5, the sum is 15 -- not a perfect square
If n=49, the sum is 1225 -- a perfect square (35²).
Can you find the first five numbers n where this happens (we have given you two of them)?

Solution to the Problem:

Here is the solution:

The first five integers n for which the sum is a perfect square are:
1, 8, 49, 288, and 1681. ​


Reacall that 1 + 2 + 3 + ... + n = (n(n+1)) / 2
So we are looking for n such that (n(n+1)) / 2 = k²