Too Square

Too Square
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Square numbers aren't the only ones that we classify by shape.

Triangular numbers: 1, 3, 6, 10, 15, 21, 28, . . .
You should build this series of numbers with the color tiles--how many patterns can you see?

Pentagonal numbers: 1, 5, 12, 22, 35, 51, 70, . . .
This pattern is impossible to build with color tiles but you should be able to figure out the pattern using these drawings:

There are some very interesting patterns in these different number series. First, let's look at the triangular numbers.

1 3 6 10 15 21 28

Now, let's figure out the difference between each of them. (Did you notice that triangular numbers are the sums of consecutive numbers?) We subtract 1 from 3 and get 2, 3 from 6 and get 3, and 6 from 10 and get 4. Interesting pattern, don't you think? These numbers are called the first differences.

1 3 6 10 15 21 28

2 3 4 5 6 7

The next step is to do the same thing with the first differences. Subtract 2 from 3 to get 1, 3 from 4 to get 1, 4 from 5 to get 1. Yikes!! The second difference for triangular numbers is always 1.

1 3 6 10 15 21 28

2 3 4 5 6 7

1 1 1 1 1

You will find a very interesting pattern if you do the same for square numbers, and pentagonal numbers. In fact, if you try all of those, it will be obvious what the second difference is for hexagonal numbers (1, 6, 15, 28, 45, 66, . . .).

Just think, before you started this page you would have thought that a question like, "What is the second difference for hexagonal numbers?" was way too hard for you. Now you know it's a cinch! Way to go!!!