Squangular Numbers and Theon’s Ladder (ESSAY!)
The list of square numbers begins 1, 4, 9, 16, 25 … and these represents the counts of pebbles that can be arranged in square arrays. The list of triangle numbers begins 1, 3, 6, 10, 15, 21, 28, 36, … and these represent the counts of pebbles that can be arranged in triangular arrays. (For example, 10 pebbles can be arranged as a row of 1, a row of 2, a row of 3, and a row of 4 making a triangle pattern of dots.)
Notice that the number 36 is both square and triangular. (As is the number 1 in a trivial way.)
What is the next “squangular” number? How many are there? Can we list them all? Is there a general formaul for them? YES!
The video is a teaser that gives away the answers. The essay is the full story of the mathematics behind the scenes. (WARNING: This is advanced work, still accessible to high-schoolers, but is serious in its efforts.)