Fun with squares

Well, I was sick yesterday, and as I was lying in bed trying to get some rest, I started thinking about squares. Don’t ask me why, I just did.

Nevertheless, did you realize that if you start with 0, and continue to add consecutive odd numbers, each sum will be the next square? This is what I mean:

0 = 0 x 0 (0 squared)

0 + 1 = 1 (1 squared)

0 + 1 + 3 = 4 (2 squared)

0 + 1 + 3 + 5 = 9 (3 squared)

0 + 1 + 3 + 5 + 7 = 16 (4 squared)

0 + 1 + 3 + 5 + 7 + 9 = 25 (5 squared)

And so on, and so on.

While contemplating this, I chanced on another observation about squares. That is, the last digit of squares follows a repeating pattern. Let me demonstrate by first listing all the squares from 0 to 20:

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400

You will notice that for each group of ten numbers (0-10, 10-20), the last digits follow a symmetrical pattern:

0 to 100 (10 squared): 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0

100 to 400 (20 squared): 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0

Obviously, this pattern continues forever and ever.

Then, as an alternative to counting sheep, I mentally continued on from 400, adding consecutive odd numbers in my head to derive each successive square: 441, 484, 529, 576, 625, and so on.

This led me to another observation. Just like a pattern exists with the last digit of a square, so also a pattern exists with the last two digits of a square. In this case, the pattern repeats every 50 numbers, and is also symmetrical. Now, instead of listing all the squares from 0 to 50, let me demonstrate by taking just the squares from 20 to 30:

400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900

Do you see how the last two digits follow a symmetrical pattern centered at 625 (25 squared)?

So even though I didn’t list all the squares, you can pick out any two “symmetrical” numbers between 0 and 50 as a check. Take 12 and 38 for instance. According to this pattern, they should have the same last two digits. And alas, it is so. Twelve squared is 144, and 38 squared is 1444.

Going one step further, since this pattern repeats with each 50 numbers, one can see that 12 squared, 38 squared, 62 squared, and 88 squared all end in “44.” Do you follow?

At this point, my head started to hurt worse than when I laid down. So while I believe that there is probably no end to the fun one could have by thinking about squares, I reluctantly decided to stop for the time being. It’s not the best activity to undertake when you are sick and trying to sleep.