It was towards the end of the school year. Teachers were winding down for the summer. Issue 17 of Chalkdust had just been published. Life was good. But maths doesn’t stop when exams stop. In fact, this is when the fun really starts.

As a teacher, I like nothing more than getting my students to challenge their mathematical intuition. We’d already debated whether maths is created or discovered (inspired by issue 17’s big argument). It was time for another challenge. I asked my students to use their mathematical intuition to determine whether they thought that there were more or less (or the same number of) even numbers than odd numbers in Pascal’s triangle.

But wait, I am getting ahead of myself. Let’s start with a brief reminder of what Pascal’s triangle is. Pascal’s work on the triangle was published in 1665, but we know that Persian mathematicians used this triangle for binomial expansion some 600 years earlier; the Chinese mathematician Jia Xian also wrote about the triangle in the 11th century; and even in Europe, Niccolo Tartaglia published the first six rows of the triangle in 1556 (in Italy, the triangle is named after him). However, it is Pascal’s work that is remembered, partly because he identified some new properties and history is oh so fickle.

The triangle is constructed row by row. The left- and right-most entries are all ones. The inside entries are then the sum of the two numbers diagonally above it:

Now we can return to the original question. Are there more or less even numbers than odd numbers, or are there the same number? Look at it and draw on your mathematical intuition: what do you reckon?

There are three possible conclusions that you might have reached:

“There are the same number of even and odd numbers.” Perhaps you’re thinking that evens and odds are in the same ratio in the set of natural numbers and so they should appear in the same ratio in the triangle.

“There are more odd numbers than even numbers.” Potentially you’ve spotted that every row starts and ends with an odd number and so you’ve surmised that that is going to skew the results in favour of the odd numbers.

“There are more even numbers than odd numbers.” This is what a lot of my students thought. Their reasoning? “Why else would you ask this question?” Since this happens to be the right answer, I guess that they know me too well. However, from the information that we currently have, this doesn’t seem like the right answer.

To gain more confidence in our answer we need more information. So, I wrote a short program that would tabulate the first twenty rows. (If you want to play with the numbers yourself, you can find my code at github.com/graeme-f/Pascal). By running the code, I was able to see that the percentage of even numbers gradually grows: by the 18th row, the total number of even numbers has exceeded the total number of odd numbers. This raised a number of questions: does the percentage of even numbers continue to increase? Does it tend to a steady value? And if so, what is that value?

I ran the code again, this time looking at more rows. By the time I had reached the hundredth row the percentage of even to odd numbers was in excess of 75%. I reached 80% on the 164th row and by the 870th row, 90% was reached.

I extended this to just over 4000 rows and by then, over 93% of the numbers were even. It was difficult to see if the trend was still growing because it wasn’t smooth, rather it came in small bumps which slowly rise and then dip down:

In order to understand the behaviour as the number of rows gets much bigger, we need to understand these bumps. The bumps can be illustrated by showing the even numbers in one colour and the odd numbers in a second colour.

This diagram has the even numbers shown by yellow shapes. Some readers may think this shape seems familiar and indeed they’d be right. The triangles here mimic those seen in the Sierpinski triangle. These inverted triangles get gradually larger and, as the triangle starts to taper off, other (smaller) triangles will appear to the side.

The diagram provides some clues as to why there are so many even numbers. If there is a row of even numbers then the next row will have one fewer even number, because an even number plus an even number will result in another even number, and the numbers on the next row are created from the sum of two numbers in the previous row. It will be one fewer because of the two edges which are always one. Conversely, if there is a row of odd numbers, then the next row will have a row of even numbers, because an odd number plus an odd number will result in an even number. Obviously, if there is an odd and an even number then that will produce an odd number, but because the edge is always one then an odd number that is generated next to the edge will become an even number in the next row. This starts to be propagated into the centre of the triangle and, as the triangle gets larger, the ones (which are odd) that appear on the edge of the triangle are less significant to the total count.

Moving down the centreline, it can be seen that there are five distinct triangles, each significantly larger than the previous one. The side length of each triangle is 1, 3, 7, 15 and 31. These are all one short of powers of two. With a little work, you can show that the size of the triangles obey the formula $2^n-1$ where $n$ is the triangle number starting from one.

So that answers our question: Pascal’s triangle has more even numbers than odd numbers. But what about multiples of other numbers? Well, I’m glad you asked. Colouring the multiples of 2 yellow gave us this pattern:

Instead, colouring the multiples of 3 yellow gives:

Where the 2-triangle had triangles of size 1, 3, 7, 15 and 31, the 3-triangle has triangles of size 2, 8 and 26. This follows a pattern of $3^n-1$. What is more, these come in groups of three. The number 3 is many things, but the property I will highlight is that it’s the 2nd triangle number.

Colouring the multiples of 5 yellow gives this:

The 5-triangle has triangles of size 4 and 24, following the pattern of $5^n-1$, and these are grouped in batches of 10—the 4th triangular number.

The multiples of 7 look this this:

The 7-triangle continues the pattern with triangles of size 6 and 48 (or $7^n-1$) and they come in batches of 21, the 6th triangular number.

In fact, all prime numbers appear to follow this regular pattern: triangles are all of size $p^n -1$, and are grouped in batches of the $(n-1)$th triangle number.

Composite numbers are, however, more fragmented. The 6- and 4-triangles illustrate two different characteristics of triangles based on composite numbers. First, let’s look at the 6-triangle:

The 6-triangle is a combination of the 2-triangle and the 3-triangle. Where these two overlap, a multiple will appear.

Now, look at the 4-triangle:

4 is a power of 2, so the 4-triangle has the same structure as the 2-triangle, but with more gaps. And yet, these gaps themselves form a regular pattern. This can be explored further by comparing additional powers of 2:

Looking at these triangles we see that, as the exponent of 2 increases, each yellow triangle bigger than one shape is split up. In fact, it’s split in a similar Sierpinski pattern as our multiple-of-2-triangle, but with the colours inverted. So, going from 2 to 4, a single triangle will disappear, a $3\times 3$ triangle becomes just 3 single triangles each at the apex, a $7\times7$ triangle becomes three $3\times 3$ triangles with a single triangle in the centre, and the $15\times 15$ triangle becomes three $7\times 7$ triangles with the centre triangle being replaced with a single $3\times 3$ and three single triangles. This same pattern is then replicated going from 4 to 8 and 8 to 16.

Now we do what mathematicians do, and check to see if a similar pattern can be seen when looking at powers of 3. And it can! The only difference is that, now, the triangles are reduced in blocks of $2\times 2$ triangles.

Returning to my students (remember them?), they were in awe. We had uncovered these patterns embedded deep within the triangle. Even more exciting was when they realised that the triangle itself was independent of any number-base system. Yet, it was able to highlight these characteristics of prime numbers. We saw a regular structure, a fingerprint if you will, of each prime number. It’s certainly not a meaningful way to find prime numbers, but it is a way to recognise them, a way that is independent of our number system, and perhaps most importantly, a beautiful way to discover more mathematics.

And so, as the curtain fell on another school year, I requested my students, as I request you now, to keep asking questions…because you will never know what is there to be discovered.