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The Hockey Stick and Parallelogram Property of Pascal’s Triangle

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

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 ten 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 some of these sums for yourself to find out. This is one of my favorite patterns in Pascal’s triangle – it’s really quite surprising that this property always seems to work, and yet, as we’ll see, it’s actually not too hard to prove!

As an example, I will show the idea behind the proof with the sum shown in the diagram above. We’ll start from the bottom of the hockey stick at 35, the total of 1, 3, 6, 10, 15, and 21. As in Pascal’s triangle, each number is the sum of the two above it, we can start by writing the sum 35 = 15+20.

Now the 15 is on the Hockey Stick row (the row of numbers in this case in the second column). But what about the number 20? Change it to a sum of the two above! We get 20 = 10 + 10, and so our overall sum becomes 35 = 15 + 10 + 10. Now we have a sum where 15 and one of the 10s is on the Hockey Stick line. We continue this process, having only one number not on the line each time, 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 that we don’t have in our sum that is on the line is also a 1. The whole process for 35 is shown below (the numbers in bold are those on the line of the hockey stick:

35 = 15+20

35 = 15+ten+10

35 = 15+10+6+4

35 = 15+10+6+3+1

So it’s clear why the Hockey Stick property of Pascal’s Triangle works, although it still makes it an interesting pattern that can also be expanded into many other patterns such as the Parallelogram property.

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