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## Main Question or Discussion Point

**"Playing" with math keeps me sharp!**

I'm pretty new to math, just taking Calc II and a proof class.

But every once in a while, while doing homework, I get obsessed with looking for patterns in math and expressing them. They are usually pointless, and come out to be kind of "duh" in the end, but man, I feel like I can do anything after doing that. It's fun to do.

Maybe one day, I'll come up with something useful. Here is a walkthrough of what I did tonight.

During Physics homework, I got "reminded" of the idea that a square has the most area per perimeter of any rectangle. Ergo, decreasing one side and increasing the other will give a smaller area. So I looked for a pattern there.

10 x 10 = 100

9 x 11 = 99

8 x 12 = 96

7 x 13 = 91

6 x 14 = 84

5 x 15 = 75

....

The numbers decrease from the next in the pattern of odd numbers, 1, 3, 5, 7, 9, (100 - 99 = 1; 99 - 96 = 3; 96 - 91 = 5)

So I looked for a pattern there, by adding up those numbers. For example, 5 x 15 is (1 + 3 + 5 + 7 + 9 = 25) away from 100. After writing all of those:

10 x 10 = 100; 0

9 x 11 = 99; 1

8 x 12 = 96; 4

7 x 13 = 91; 9

6 x 14 = 84; 16

5 x 15 = 75; 25

...

I quickly realized that those numbers (1, 4, 9, 16) are the difference that either of the terms are from 10, squared. So 8 x 12 is 4 away from 100, and 12 and 8 are 2 away from 10, then square that and you get 4.

So I expanded it to say that those numbers just average to a certain number, in this case 10. So, I then noted that multiplication of two numbers is equal to the average of the two numbers, squared, minus the difference either number is from the average, squared!

Or written out:

http://img28.imageshack.us/img28/1725/conjecture01.png [Broken]

Tomorrow I'm going to do something with cubes, AKA ABC.

Do you guys "play" with math in this way?

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