I like math. It’s not a phrase you’ll hear people say a lot, but there it is: I like math. If my brain feels fuzzy, I’ll go looking for online algebra problems to solve. When I take walks, I try to figure out the relationships between the integers that make up house numbers. My longtime favorite “game” is answering the question, “can I create all the numbers from 1-10 using only the numbers on this house and basic arithmetic functions? If the house number has 4 or more digits to it, the answer is usually, “yes.”
Recently, I came up with a new game: find the largest number that can be made from the integers on this house. At first I was just using basic arithmetical functions. But then I started using exponents–whee!
Okay, that last sentence makes me sad for me, and I’m the one who wrote it. That being said, the numbers get so huge so quickly, I can’t help but get excited. There’s something humorous about absurdly large numbers that just makes me grin.
Anyway, on Monday, I started playing with the numbers: 2 2 5 8. I wanted to get the biggest number possible, so I put these integers into this equation: 2^2^5^8*. Then, to get started on the math, I begain by calculating the powers of five in my head. As I did so, I noticed two patterns. Take a look:

5^8= 5x5x5x5x5x5x5x5
=(25)5x5x5x5x5x5
=(125)5x5x5x5x5
=(625)5x5x5x5
=(3,125)5x5x5
=(15,625)5×5
=(78,125)5
=390,625

The two things I noticed are:
1) the last two digits of any power of five are always 25 and:
2) the last three digits appear to switch between 125 and 625 after 5^3.

I’m sure that real mathematicians would roll their eyes and tell me how disgustingly obvious this all is. But I hadn’t noticed it before, and I thought it was cool.
Incidentally, when I tried plugging in 2^390,625 into a calculator, it told me the answer was: INFINITY. Other calculators basically told me that the number was too big to calculate. Or, in street terms: AHHH! You’ve broken my mind!
So, does anyone else play with numbers this way? And is there some term for patterns like this in powers of numbers?