After much thought, I've hit upon a math puzzle that will work for my friend's little neighbor, the aspiring mathematician. Actually, I'm going to give her a two-part puzzle: a more straightforward version of the "littler math puzzle" posted below, and this little guy that I came up with yesterday while exploring the behavioral sciences building. Incidentally, why do "people scientists" and "medical scientists" always have such nicer buildings than "hard scientists?"
Anyway, here's the puzzle:
Most small numbers are pretty close to a prime number. For example, the number 22 isn't prime, but if I just change the second 2 into a 9, then it is (because 29 is prime). Similarly, 310 isn't prime, but 313 is. So it seems like often, all I have to do is change one digit in a number and I can make it into a prime number.
Unfortunately, that's not true for ALL numbers. There are some numbers that can't be made into a prime number just by changing one (and only one) digit. The question is, what is the smallest of these numbers?
Answer and explanation below the fold, as always.
The answer is "200."
I really like this puzzle, for some reason. I dunno why; it's quite straightforward, but somehow it feels rewarding to figure it out even if it's not particularly hard to do so. Anyway, I don't actually know of any real "trick" to the puzzle; I think you just have to start at the beginning and go up until you find one. The key is going up in a smart way.
It can't be a one-digit number, for obvious reasons. So maybe it's a two-digit number? It's not hard to see that it can't be a two-digit number either. Why? Because I can think of a prime number in each decade worth of two-digit numbers; for example, 11, 23, 37, 41, 59, 67, 73, 83, and 97. So if you give me a two-digit number, no matter what the first digit is, I can change the second digit so that it matches one of these.
Okay, so maybe it's a three-digit number. Could it be a three digit number in the 100's? I don't think so. But don't worry, we don't have to check all 99. I claim that, if you want to construct a number that can't be made prime just by changing one digit, then you want it to end with a zero. That way, if anyone tries to change the second or third digit (the tens' or hundreds' place) then they'll still be stuck with a number that's divisible by 10, and therefore not prime. Obviously, they could do the same thing by making it end in 2, or 4, or 5, etc., but since we're looking for the smallest number like this then we should use a zero.
Alright, so we think the number might look like this:
1x0
where x is some digit. But now I am going to pull the same trick I pulled before to prove that it can't look like this, because I can find prime numbers for each decade; namely, 101, 113, 127, 137, 149, 151, 163, 173, 181, and 191. So no matter what number you choose for "x," I have a way to change the last digit and make the whole thing prime.
So alright, lets check the 200s. As before, we probably want something that looks like "2x0" for some number x. Can I once again find a prime number in each decade? No, I can't. In fact, I can't do it right off the bat. There are no prime numbers between 200 and 210. That means if I pick "x" to be a zero, there should be no way to change just one digit and get a prime number.
Let's just check that this is true. Consider "200," and imagine changing the first digit. Whatever you get will be divisible by 100, so obviously it's not prime. Now imagine changing the second digit; the result will always be divisible by 10. So Finally, maybe I can change the third digit. But I just argued (check if you don't believe me) that there are no prime numbers between 200 and 210, so no matter what I try (207, 204, etc.) I will not be able to do it.
What do you think? I kind of like it. And in fact, I've enjoyed coming up with these little math puzzles so much, I think I may try to post one on a regular basis. Not so frequently as these first few, of course, but maybe every week or two?
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