Squares, triangles, and octal

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 (2n + 1)².

Both parts of the theorem above follow from the calculation

( (2n + 1)² − 1 ) / 8 = n(n + 1) / 2.

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

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