I ran across the following theorem in Ross Honsberger’s book Mathematical Morsels:

Every odd square ends in 1 in base 8, and if you cut off the 1 you have a triangular number.

A number is an odd square if and only if it is the square of an odd number, so odd squares have the form (2*n* + 1)².

Both parts of the theorem above follow from the calculation

( (2*n* + 1)² − 1 ) / 8 = *n*(*n* + 1) / 2.

In fact, we can strengthen the theorem. Not only does writing the *n*th odd square in base 8 and chopping off the final digit give *some* triangular number, it gives the *n*th triangular number.