Patterns in Pascal’s Triangle

One of the great joys in Maths is exploring something seemingly very simple and finding layers upon layers of complexity, and connections with other areas of Maths which at first sight appear to be totally separate. I’m sure you have come across Pascal’s triangle; here it is:

To create each new row, start and finish with 1, and then each number in between is formed by adding the two numbers immediately above.

Pattern 1:   One of the most obvious patterns is the symmetrical nature of the triangle. It’s fairly obvious why: underneath 1 2 1 there must be 3 3 (because of the 1 + 2 and 2 + 1), and the symmetry carries on from there.

Pattern 2:   Another obvious pattern appears down the second diagonal (either from left or right) which forms the counting numbers.

Again, the reason is obvious: having created 2 as 1 + 1, 3 appears as 1 + 2, then 4 = 1 + 3 and so on. Will the pattern carry on for ever? Yes, it must do.

Pattern 3:   Now look at the next diagonal, and you should recognise the sequence 1, 3, 6, 10 …

These are the triangle numbers. Can you see how they are formed?

3 = 1 + 2
6 = 1 + 2 + 3
10 = 1 + 2 + 3 + 4

So, for example, the 6 is formed from 3 + 3 in the row above, and the left hand 3 = 1 + 2. Thus we get 6 as 1 + 2 + 3. Now add the 4 to the left of the 6 to get 10: 6 + 4 = (1 + 2 + 3) + 4, and so on.

Pattern 4:   Here’s one you can’t see without doing a simple calculation. Add all the numbers in each row, and what pattern do you get?

Row 0:   Sum = 1
Row 1:   Sum = 2
Row 2:   Sum = 4
Row 3:   Sum = 8

Now check that rows 4 and 5 give 16 and 32. By convention, the top row is known as the 0th row, the next row is the 1st row and so on. (One reason for this is that the 3rd row begins 1 3, the 4th row 1 4 and so on.) Thus the sum of the numbers in the nth row is 2ⁿ. The proof of this is a bit more complicated, but you can easily look it up if you want to.

Pattern 5:   While we’re on powers, the digits of each row form powers of 11. Row 0 = 1 = 11⁰; row 1 = 11 = 11¹; row 2 = 121 = 11²; row 3 = 1331 = 11³. But what do we do with the double digit numbers in row 5? Well, wherever there is a double digit number, add the left hand digit to the number in the previous cell. Thus the digits in row 5 become: 1, 5 + 1, 0 + 1, 0, 5, 1 which gives 161051 = 11⁵. I love it!

Pattern 6:   Here’s a weird one. If we shade the odd and even numbers in different colours we get …

The pattern we get is the start of the Sierpinski Triangle. This is formed as follows:

There are dozens more patterns hidden in Pascal’s triangle. Further, the numbers themselves have all sorts of uses, and you may have come across some of them in areas such as probability and the binomial expansion. Blaise Pascal discovered many of its properties, and wrote about them in a treatise of 1654. However, it appears that the triangle was known about at least as far back as the 11th century when both Persian and Chinese mathematicians were working on it independently.