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    • The hidden beauty of multiplication tables

      Zoheir Barka
      7 April, 2017
      7 comments

      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.

      Table 0

      We will add a row of 0s at the top and a column of 0s on the left. This still gives a consistent table — the first row and column contain multiples of 0,  the second row and column contain multiples of 1,  the third row and column contain multiples of 2, etc — and it will provide a nice frame for our patterns.
      Table 1

      In the following, we will colour the squares of the multiplication table that correspond to multiples of a number k for various values of k. And we'll discover some beautiful symmetries.

      Single multiples

      We begin with k=2: we assign the colour blue to every square in the multiplication table that is a multiple of 2. (The number 0 is a multiple of 2, so all the 0 squares are blue.)
      Table 2

      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:

      Fundamental building block

      The fundamental building block contains k×k=2×2=4 cells of the multiplication table. The squares defined by the white cells in the pattern consist of (k−1)2=(2−1)2=1 cells. Below are two more images in which the multiples of a number k have been coloured blue. Can you tell what the value of k 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.
      Table 3

      Table 4

      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 2 are coloured red, and those that are multiples of 3 are coloured orange (with the orange taking precedence over the red in the case of multiples of both 2 and 3, that is, multiples of 6).

      This gives the following pattern.

      Table 5

      Note that this time our fundamental building blocks consist of 6×6=36 little squares, which makes sense, because 6 is the least common multiple of 2 and 3. The symmetry emerges from repeated copies of a 5×5 square with a nice four-fold symmetry. The next figure takes this a step further, assigning red to numbers that are multiples of 2, orange to numbers that are multiples of 3, and yellow to numbers that are multiples of 4. If a cell is a multiple of two of these numbers (eg 6=2×3) then it will be assigned the colour of the larger of these two numbers (orange in the example). We will stick to this convention for the rest of this article.
      Table 6

      This time, our fundamental building blocks contain Math input errorMath input error cells, which again makes sense, given that 12 is the least common multiple of 2, 3 and 4. The symmetry emerges from repeated copies of an Math input errorMath input error square, which contains nine small Math input errorMath input error 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.)

      Table 7

      Table 8

      Table 9

      Table 10

      Multiples of non-consecutive numbers

      Next we use some non-consecutive values of k. The following figure uses blue for numbers that are multiples of 6, and green for numbers that are multiples of 9.

      (Click on the image to see a larger version.)

      Table 11

      The fundamental building blocks will now consist of 18×18=324 little squares, as 18 is the least common multiple of 6 and 9. Still the additional symmetries within the nine 5×5 squares that make up the repeated 17×17 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.)

      Table 12

      Table 13

      Table 14

      Table 15

      Table 16

      Remainders

      Finally, if we fix a number k and assign colours to cells depending on their remainder with respect to k, then all the squares can be filled in. For example, let multiples of 5 be black, numbers with a remainder 1 with respect to 5 be green, numbers with a remainder 2 with respect to 5 be red, numbers with a remainder 3 with respect to 5 be purple, and numbers with a remainder 4 with respect to 5 be yellow; the following figure is obtained.:

      (Click on the image to see a larger version.)

      Table 17

      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.

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      Comments

      James S.

      11 April 2017

      Permalink

      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.

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      Marianne

      11 April 2017

      In reply to Typo in last pattern... by James S.

      Permalink

      Thanks for pointing out the mistake, we have corrected it.

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      Raúl A. Simón Eléxpuru

      14 April 2017

      Permalink

      ¡Beautiful! Congrats to the author.

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      Oli

      19 April 2017

      Permalink

      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!

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      Barka

      25 April 2017

      In reply to Animated? by Oli

      Permalink

      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/

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      Ken Wessen

      27 June 2017

      Permalink

      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…

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      Barka

      4 February 2025

      Permalink

      In previous work, we have explored some of the symmetries hidden in the multiplication tables of natural numbers and integers. In this article, we dive deeper into the hidden symmetries within the distribution of positive and negative integers. We also share various ideas to explore these symmetries in a classroom setting, encouraging active engagement and deeper understanding among students.

      https://scholarship.claremont.edu/jhm/vol15/iss1/17/

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