April 2026
Problem of the Month

Triangular Numbers
Submitted by Davit Banana of Turkey

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:

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

Correctly solved by:

1. Colin (Yowie) Bowey	Beechworth, Victoria, Australia
2. Kamal Lohia	Hisar, Haryana, India
3. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
4. Ivy Joseph	Pune, Maharashtra, India
5. Dakoda Perrigo	Central High School, Grand Junction, Colorado, USA
6. Kelly Stubblefield	Mobile, Alabama, USA
7. Sage Siegrist	Central High School, Grand Junction, Colorado, USA

Send any comments or questions to: David Pleacher