Sequences and PatternsPascal’s Triangle

Below you can see a number pyramid that is created using a simple pattern: it starts with a single “1” at the top, and every following cell is the sum of the two cells directly above. Hover over some of the cells to see how they are calculated, and then fill in the missing ones:

This diagram only showed the first twelve rows, but we could continue forever, adding new rows at the bottom. Notice that the triangle is symmetricright-angledequilateral, which can help you calculate some of the cells.

The triangle is called Pascal’s triangle, named after the French mathematician Blaise Pascal. He was one of the first European mathematicians to investigate its patterns and properties, but it was known to other civilisations many centuries earlier:

Pascal’s triangle can be created using a very simple pattern, but it is filled with surprising patterns and properties. That’s why it has fascinated mathematicians across the world, for hundreds of years.

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Finding Sequences

In the previous sections you saw countless different mathematical sequences. It turns out that many of them can also be found in Pascal’s triangle:

Of course, each of these patterns has a mathematical reason that explains why it appears. Maybe you can find some of them!

Another question you might ask is how often a number appears in Pascal’s triangle. Clearly there are infinitely many 1s, one 2, and every other number appears at least twiceat least onceexactly twice, in the second diagonal on either side.

Some numbers in the middle of the triangle also appear three or four times. There are even a few that appear six times: you can see both 120 and 3003 four times in the triangle above, and they’ll appear two more times each in rows 120 and 3003.

Since 3003 is a triangle number, it actually appears two more times in the third diagonals of the triangle – that makes eight occurrences in total.

It is unknown if there are any other numbers that appear eight times in the triangle, or if there are numbers that appear more than eight times. The American mathematician David Singmaster hypothesised that there is a fixed limit on how often numbers can appear in Pascal’s triangle – but it hasn’t been proven yet.

Divisibility

Some patterns in Pascal’s triangle are not quite as easy to detect. In the diagram below, highlight all the cells that are even:

It looks like the even number in Pascal’s triangle form another, smaller trianglematrixsquare.

Colouring each cell manually takes a long time, but here you can see what happens if you would do this for many more rows. And what about cells divisible by other numbers?

Binomial Coefficients

There is one more important property of Pascal’s triangle that we need to talk about. To understand it, we will try to solve the same problem with two completely different methods, and then see how they are related.

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