The Hockey Stick and Parallelogram Property of Pascal’s Triangle

The “Hockey Stick” property states that the sum of any diagonal line starting from a 1 on the outside of the triangle is the number diagonally down from the last number, in a hockey stick shape. When the numbers of Pascal’s triangle are left justified, this means that if you pick a number in Pascal’s triangle and go one to the left and sum all numbers in that column up to that number, you get your original number. This sounds very complicated, but it can be explained more clearly by the example in the diagram below:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1+3+6+10+15+21 = 35

Try a couple of these sums out for yourself to get the hang of them. This is one of my favourite patterns in Pascal’s triangle – it really it quite a surprising that this property seems to always work, and yet, as we are about to see, it is actually not too hard to prove!

As an example, I am going to shown the idea behind the proof with the sum shown in the diagram above. We will start with the bottom of the Hockey Stick at 35, the total of the 1,3,6,10,15 and 21. As in Pascal’s triangle every number is the sum of the two above it, we can start by writing the sum 35 = 15+20.

Now, the 15 lies on the Hockey Stick line (the line of numbers in this case in the second column). But what can we do about the number 20? Change it into a sum of the two above! We get 20 = 10+10, and so our overall sum becomes 35 = 15 +10+10. We now have a sum where both 15 and one of the 10s lie on the Hockey Stick line. We continue this process, each time having only one number not on the line, until we reach the edge of the triangle, where our number not on the line is a 1. Then, we are done because the remaining number we haven’t got in our sum which is on the line is also a 1. The whole process for 35 is shown below (the numbers in bold are the ones which lie on the hockey stick line:

35 = 15+20

35 = 15+10+10

35 = 15+10+6+4

35 = 15+10+6+3+1

It is clear, therefore, why the Hockey Stick property of Pascal’s Triangle works, although this makes it no less an interesting pattern which can also be developed into many other patterns such as the Parallelogram property.