Table patterns

Neeraj Naidu

A mathematician, like a painter or poet, is a maker of patterns. – G.H.Hardy

Cramming multiplication tables is an utterly meaningless task. In a primary level mathematics class, children singing multiplication tables like a song is a common sight. Math teachers not only insist but force children to memorize the tables at least up to 20.

Well, it definitely speeds up calculations and thus helps solve textbook problems quickly. But what is the point? Calculation is not mathematics at all. It is a fragment of a fragment of a fragment perhaps.

First thing that we can do in a math classroom is to clear the air and show children that there is no need to hopelessly memorize the tables. If you can add numbers, you can very well multiply because multiplication is just repetitive addition.

However, let’s not reject the tables entirely. They could just prove to be a perfect tool to find and see beautiful patterns, something inherent to mathematics.

Let’s take an example:

What do you SEE?

The standard (numbers being multiplied to numbers from 1 to 10) multiplication table of 9, 19, and 29, right?
Perfect. But can you see anything else in the multiplication tables?
You can, can’t you? It’s because we all have a gift to see patterns, the great gift of mathematics.

Some of the things you might have observed are:
• All three numbers (9, 19, and 29) have 9 in their units place.
• The common difference between the numbers is 10: 29-19=19-9=10.
• Repetitive addition: 9+9=18, 18+9=27, 27+9=36… 19+19=38, 38+19=57….
• 29-19=19-9=10, 58-38=38-18=10×2=20, 87-57=57-27=10×3=30….The difference goes on like: 10, 20, 30, 40, 50, 60, … as the multiplication progresses.
• When 9, 19, and 29 are multiplied with an odd number, the result is always odd and when multiplied with an even number, the result is always even. Does this mean odd x odd = odd and odd x even = even? Can we generalize just by the results we have? Or do we need more numbers to prove it? Or can there be any method to prove this conjecture?
• Anything more?

Let me write the same tables with some space between the digits.

What do you think now?

You must have remembered or perhaps observed this trick that the multiplication table of 9 does. The unit place digits decrease – 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.
Is it the same case with 19 and 29?
Oh yes! In both the numbers, the results have unit place decreasing – 9, 8, 7, 6, 5, 4, 3, 2, 1, 0. Do you think 39, 49, and 59 will also follow this pattern?
Yes? So, can we make an assumption that the numbers in the unit place decrease by 1 each time? Let’s call this Assumption 1.
Now looking at the results, what do you think about the numbers written on the left of the unit place. Do you see any pattern?
Yes? In the table of 9, can you see the numbers go up? 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? So, can we say that the numbers increase by 1 each time? Let’s call this Assumption 2.
Similarly, in the table of 19, you can see this pattern: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. Each time the numbers increase by 2. And thus, following the pattern, in the table of 29 the numbers should increase by 3 each time: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.

Is it like this:

Now that you have made some assumptions or let’s say established some patterns, let’s see what happens when we do some more multiplication.

Does the pattern we have established break anywhere?

Does this mean both the assumptions we made are wrong?

The numbers in the unit place stop decreasing by 1 when 9, 19, and 29 are multiplied with 11. Instead, the pattern 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 is repeated again. This falsifies Assumption 1.

Let’s talk about Assumption 2.
Can you see that our assumption of the increase in number (left to unit place) has not been followed when we multiply the numbers with 11? However, the pattern is broken in a sort of pattern again. In the table of 9 the increase has dropped from 1 to 0, in 19, the drop is from 2 to 1 and in 29, the drop is from 3 to 2, which means in all three numbers there is a decrease of 1.
Strangely after multiplication with 11, all the results follow the pattern according to Assumption 2.
Does this mean Assumption 2 is also invalid?
Is there a reason why the pattern breaks? Or there is actually no pattern and we are pointlessly stressing on finding them?
Think. Can we trust mathematics?
Does the pattern really break? Or it doesn’t and we have made a silly mistake?
Yes. The mistake is in our observation. (It is also possible that you did not make the mistake and made the correct observation. I am just trying to highlight the case of making this mistake, which usually happens in my classroom. Anyway, mistakes are a good thing.)
What is on the left of the unit place?
The tens place, right?
Shouldn’t the pattern be like:

Do you think Assumption 1 we’ve made earlier and falsified later is actually not incorrect? The numbers in the unit place keep decreasing by 1. We can definitely decrease from zero and move to negative numbers.
Now in the tens place, the numbers keep increasing not by 1 but by 10. For example: in 43, the ten’s place digit is 4 and it means four tens which means 40.
Similarly, for 19, the increase in tens place is by 20. And for 29, the increase in tens place is by 30.
Hence Assumption 2 is incorrect and now we can frame a new pattern that might not break.
I hope this pattern doesn’t break. It doesn’t seem like it. But how could we know for sure? We can’t keep multiplying till eternity.
I hope you’ll find more patterns in this. Let me finish this particular example with one more beautiful pattern.

This next one is for you.

See and find patterns. Are there any striking similarities and dissimilarities in the 11-21-31 series and the 9-19-29 series? Can you make more examples like this that brings the hidden art of mathematics to light?

The author works with children in and out of schools. He is interested in children’s literature, mathematics, libraries, and radical pedagogies. He can be reached at irockmad@gmail.com.