Maybe it’s because I’m in the midst of writing hundreds of problems for SUMiT 2012 that I feel like I’m seeing palindromic numbers everywhere! (Even in the date: 12/21…)
For those of us born in or before 1991, we’ve lived through two palindromic years: 1991 and 2002, and there won’t be another palindromic year for another century.
But did you ever happen to notice that if you take 2011, which isn’t a palindrome, and multiply it by its reversal, 1102, you get a palindrome? .
That didn’t happen in 2010: , nor the year before that: .
But…it happens again next year: .
So, we’re about to live through two consecutive non-palindromic years whose product with their reversals is palindromic…something that won’t happen again until…any guesses?
It won’t happen again until the years 10011 and 10012! That’s a long time to wait.
This palindrome madness is all a bit silly anyway…after all, if you switch to, say, base 16, the palindromes go poof! (And if you play this game with the coefficients of a polynomial, the product will always be palindromic.) So let’s not attach undue significance to these pieces of mathematical trivia…it’s all in the lightness of the holiday spirit.
p.s. If you’d like a math problem to think about, here’s one: Suppose you pick a random whole number between 1 and 1,000,000 (inclusive) with all choices equally likely. What is the probability that the number is a palindrome? More generally, if the whole number is selected from between 1 and N, roughly how does the probability depend on N?