In this article we explore some of the symmetries that hide within the multiplication table of positive whole numbers.
Let us start with the standard multiplication table. The table below contains the numbers 1 to 10 in the first row and the first column. Any other square contains the product of the first number in its row and the first number in its column.







In the following, we will colour the squares of the multiplication table that correspond to multiples of a number for various values of
. And we’ll discover some beautiful symmetries.
Single multiples
We begin with





Here we have extended the table a bit so that it runs until the number 15 in the horizontal direction. Indeed, since the complete multiplication table on positive integers is infinite on two sides, we will continue to tweak the dimensions of the tables in what follows to display the emerging patterns more clearly.
Note that the whole pattern above can be pieced together using the fundamental building block:

The fundamental building block contains cells of the multiplication table. The squares defined by the white cells in the pattern consist of
![]() |
cells.
Below are two more images in which the multiples of a number have been coloured blue. Can you tell what the value of
is in each case? Can you tell what the fundamental building blocks are, how many cells they contain, and how many cells make up the squares defined by the white cells? You can post your answers in the comment field below — in case you can’t work them out, we’ll publish the answers in a few weeks’ time.


Multiple multiples of consecutive numbers
A more interesting pattern emerges if we use multiple multiples, and corresponding to them, multiple colours. In the following figure, the numbers that are multiples of




This gives the following pattern.

Note that this time our fundamental building blocks consist of little squares, which makes sense, because
is the least common multiple of
and
. The symmetry emerges from repeated copies of a
square with a nice four-fold symmetry.





This time, our fundamental building blocks contain cells, which again makes sense, given that
is the least common multiple of
,
and
. The symmetry emerges from repeated copies of an
square, which contains nine small
squares which together create a nice four-fold symmetry.
We can carry on playing this game indefinitely. The next four figures use multiples of four, five, six and seven consecutive numbers respectively, and four, five, six and seven colours respectively. What patterns can you discern? Can you find any axes of (reflectional) symmetry? What should be the size of the fundamental (repeating) building blocks of symmetry in each case? Remember that you can post your answers in the comment field below, and in case you can't work them out, we'll publish the answers in a few weeks' time.
(Click on the images to see a larger version.)




Multiples of non-consecutive numbers
Next we use some non-consecutive values of . The following figure uses blue for numbers that are multiples of
, and green for numbers that are multiples of
.
(Click on the image to see a larger version.)

The fundamental building blocks will now consist of little squares, as
is the least common multiple of
and
. Still the additional symmetries within the nine
squares that make up the repeated
squares may come as pleasant surprises. Can you find mathematical explanations for these?
Here are a few more patterns for you to admire. In each case we colour the multiples of non-consecutive numbers. Can you tell what numbers these are and describe the patterns that emerge? Remember that you can post your answers in the comment field below, and in case you can't work them out, we'll publish the answers in a few weeks' time.
(Click on the images to see a larger version.)





Remainders
Finally, if we fix a number and assign colours to cells depending on their remainder with respect to
, then all the squares can be filled in. For example, let multiples of
be black, numbers with a remainder
with respect to
be green, numbers with a remainder
with respect to
be red, numbers with a remainder
with respect to
be purple, and numbers with a remainder
with respect to
be yellow; the following figure is obtained.:
(Click on the image to see a larger version.)

Conclusion
We've discovered some of the symmetries that hide within the multiplication table of positive integers. It's easy to create these patterns (for example, using Excel) and they can all be explained without much difficulty using the arithmetic of whole numbers and divisibility criteria. Displaying these symmetries using colours introduces a new facet to the maths. These images and others created in similar ways may appeal to students of mathematics and the arts, and may lead to new collaborations. At the very least such images may, we hope, intrigue, amaze, and inspire.
About this article
Zoheir Barka , from Laghouat in Algeria, is an amateur and self-educated mathematician. He has a Masters degree in French language from Laghouat University and is currently a French teacher in elementary school.
A version of this article first appeared as The hidden symmetries of the multiplication table in the Journal of Humanistic Mathematics, Volume 7, Issue 1 (January 2017), pages 189-203.
Comments
Typo in last pattern...
It should say remainder 4 with respect to 5 be yellow; the following figure is obtained
numbers with a remainder 1with respect to 5 be red; the following figure is obtained.
Thanks for pointing out the
Thanks for pointing out the mistake, we have corrected it.
current article
¡Beautiful! Congrats to the author.
Animated?
These are lovely, I wonder if there's any interesting animations to be had as the integers created the patterns are changed? I suspect there are!
Hi! you would like to check
Hi! you would like to check this link, you will find an interactif tool to display the different modulo's :
http://guzintamath.com/blog/2017/02/modulus-hidden-symmetries/
Web app for patterns
Thanks for this article. It inspired me to make a web app to explore the patterns - you can find it here: http://thewessens.net/ClassroomApps/Main/multiples.html?topic=number&id…