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.)

Squangular Numbers_VIDEO

Theon’s Ladder and Squangular Numbers_ESSAY


    QUADRATICS, Permutations and Combinations, EXPLODING DOTS, and more!

    Written for educators - and their students too! - this website, slowly growing, takes all the content Tanton has developed in his books, videos, and workshops, and organizes it into short, self-contained, and complete, curriculum units proving that mathematics, at all points of the school curriculum, can be joyous, fresh, innovative, rich, deep-thinking, and devoid of any rote doing! Let's teach generations of students to be self-reliant thinkers, willing to flail and to use their common sense to "nut their way" through challenges, to assess and judge results, and to adjust actions to find success. (Great life skills!)

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