More fun with squares

Since I blogged about squares a few weeks ago, I got to thinking a little more about perfect squares, as well as the comments posted on that entry. I am coming more and more to the conclusion that numerical patterns are “beautiful” in a sense, or at least reveal some aspect of the intelligent design of God. Here’s a quick example of what I mean.

You will recall that I made three observations about perfect squares while trying to fall asleep one night. First, if you keep adding consecutive odd numbers, you will arrive at successive squares. Observe:

1 + 3 + 5 + 7 = 16 (4 squared)
1 + 3 + 5 + 7 + 9 = 25 (5 squared)
1 + 3 + 5 + 7 + 9 + 11 = 36 (6 squared)

Second, I observed that the last digits of perfect squares follow a pattern that repeats with every ten numbers. Observe the last digit of the squares of the numbers 0 through 10. Do you see the symmetry?

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0

Third, I observed that the last two digits of perfect squares follow a pattern that repeats with every 50 numbers. You will recall that I listed the squares of 20 through 30 to illustrate the pattern and symmetry of the last two digits (centered around 25 squared, or 625):

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

But all this got me to thinking. If the last digits follow a pattern, and the last two digits follow a pattern, then do the last three digits also follow a pattern? How about the last four? If so, we might find that there are really an infinite number of patterns associated with the last digits of squares. And, I would add, we might also start to see where the allegations of beauty and intelligent design originate.

Well, lo and behold, I deduced that there is in fact a similar pattern with the last three digits of perfect squares. The pattern repeats every 500 numbers, and thus the symmetry is centered around the square of 250. I’ll spare you the details of how I figured this out, but you can test it by looking at the squares of any two numbers that are “symmetrical” to 250. For instance:

274 squared (274 is 24 more than 250) = 75,076
226 squared (226 is 24 less than 250) = 51,076

And to make a long story short, from here, one can further deduce that the last four digits of squares repeat (every 5,000 numbers centered around 2,500 squared), the last five digits (every 50,000 numbers centered around 25,000 squared), and so on, and so on ad infinitum (until infinity). You have to admit that this is at least a little bit cool. Just to satisfy any lingering curiosity, let me give you one more example to show the symmetrical pattern of the last five digits of perfect squares (again, centered on 25,000 squared):

27,842 squared (2,842 more than 25,000) = 775,176,964
22,158 squared (2,842 less than 25,000) = 490,976,964

And alas, the last five digits are the same.

I have not done a lot of reading on the ancient Greek philosophers, many of whom were accomplished mathematicians (like Pythagoras). I wonder if this is the kind of stuff that they observed in their study of numbers that led them to contemplate the nature and essence of the universe—and made them renowned philosophers.