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 cells 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
The triangle is called
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.
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:
The numbers in the first diagonal on either side are all
The numbers in the second diagonal on either side are the
The numbers in the third diagonal on either side are the
The numbers in the fourth diagonal are the
If you add up all the numbers in a row, their sums form another sequence: the
In every row that has a prime number in its second cell, all following numbers are
The diagram above highlights the “shallow” diagonals in different colours. If we add up the numbers in every diagonal, we get the
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
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
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 numbers that appear more than eight times. The American mathematician
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
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?
Wow! The coloured cells always appear in
If we continue the pattern of cells divisible by 2, we get one that is very similar to the Sierpinski triangle on the right. Shapes like this, which consist of a simple pattern that seems to continue forever while getting smaller and smaller, are called
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.