Modulus and Hidden Symmetries


A really nice research paper, titled The Hidden Symmetries of the Multiplication Table was posted over in the Math Ed Community yesterday. The key ideas in the article center around (a) the standard multiplication table—with a row of numbers at the top, a column of numbers down the left, and the products of those numbers in the body of the table, and (b) modulus. In particular, what patterns emerge in the standard multiplication table when products are colored by equivalence to \(\mathtt{n \bmod k}\) as \(\mathtt{k}\) is varied?

The little interactive tool below shows a large multiplication table (you can figure out the dimensions), which starts by coloring those products which are equivalent to \(\mathtt{0 \bmod 12}\), meaning those products which, when divided by 12 give a remainder of zero (in other words, multiples of 12).


When you vary \(\mathtt{k}\), you can see some other pretty cool patterns (broken up occasionally by the boring patterns produced by primes). Observing the patterns produced by varying the remainder, \(\mathtt{n}\), is left as an exercise for the reader (and me).

Incidentally, I’ve wired up the “u” and “d” keys, for “up” and “down.” Just click in one of the boxes and press the “u” or “d” key to vary \(\mathtt{k}\) or \(\mathtt{n}\) without having to retype and press Return every time. And definitely go look at the paper linked above. They’ve got some other beautiful images and interesting questions.


Barka, Z. (2017). The Hidden Symmetries of the Multiplication Table Journal of Humanistic Mathematics, 7 (1), 189-203 DOI: 10.5642/jhummath.201701.15

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Josh Fisher

Instructional designer and editor for K-12 mathematics. My research interests center mostly around mathematics education.

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