I'm a night person, and 1:21am is my favorite time. Why? Well, if you read it as decimal 121 it's a perfect square, and if you add up the number of minutes past midnight that it represents (81) it's still a perfect square. Yeah, I know, pretty darn cool.
But it gets even better. A few nights ago as I was about to go to sleep I looked at the clock, which just happened to be hitting 1:21 at that moment. I started thinking about the number and realized that 121 isn't just a perfect square in decimal; it's also a perfect square in base 8, and base 11, and base 20, and base 7, and so on. Now this may be old hat to everyone else, but it was news to me. So is it a perfect square in every base (in which it's a valid number), I wondered? Sure enough, it is, and I realized why: for every base b, (b+1)2 = 1b2 + 2b + 1 (which is, by definition, 121 in that number system). Which also means that the square root of 121 for any base b is always 11 (i.e. b+1). How cool is that?
Putting it in those terms made me realize that the same is true of a number of 3-digit perfect squares that can be represented in decimal, like 144 (b+2)2 and 961 (3b+1)2. And also that it goes beyond 961 if you don't constrain yourself to decimal—but I'm biased against digits beyond 9, so it doesn't excite me quite as much to know that 4GG is a perfect square for every base greater than 16. And I realized that it probably applied to many squares with more than three digits as well (in fact an infinite number of them), but at that point I fell asleep.
Knowing that there are an infinite number of similar perfect squares doesn't make me admire 121 any less, though. It's not my favorite number overall, but still, I think you'd have to agree that it's got a lot going for it.
MATH PHUN: Using a 24-hour clock, there's one other time of day besides 01:21 that's a perfect square both as a decimal number and in terms of its total minutes past midnight (not counting the cheater squares between 00:00 and 00:49). The first one to find it wins a prize!*
* WARNING: Prize may suck.
20:25
Posted by: Save the Oocytes | Wednesday, February 20, 2008 at 12:57 AM
Congratulations, you've won! And the prize is...public recognition of your achievement, with a minimum of two exclamation points!
Hey, I warned you.
I'm curious: what was the method you used? (Assuming you didn't just look at it and immediately say, that's easy—20:25.)
Posted by: John Caruso | Wednesday, February 20, 2008 at 09:19 AM
How'd you?
If I were you I'd just write a program and have the answer in a few minutes. But no, I read this post during a text conversation where the other side usually took about half a minute to respond, so I just happily squared numbers while I was waiting for responses, using the most primitive method imaginable. I wish I'd started at 49, though.
And this brilliance is my justification in hoping to someday become a 3-manifold theorist.
Posted by: Save the Oocytes | Wednesday, February 20, 2008 at 02:21 PM
3-manifold theory? Sounds like you're serious about this stuff. I'm a mere dabbler.
I did the same thing—just kept squaring the numbers, calculating the minutes since midnight, and figuring out if it was still a perfect square. Just before I posted it I wrote a Perl script to perform the same algorithm, though, half out of curiosity and half to make sure I wasn't about to make an ass of myself.
Posted by: John Caruso | Wednesday, February 20, 2008 at 02:54 PM
You know, I remember a similar discussion over at Red State...
Posted by: SteveB | Wednesday, February 20, 2008 at 04:45 PM
I'm a "PhD candidate."
SteveB, about numbers?
Posted by: Save the Oocytes | Wednesday, February 20, 2008 at 08:20 PM