# Primes vs Palindromes — A Counting Puzzle

Part of a typical math journey in the school curriculum is learning about different types of numbers and their properties. These are things like even numbers, odd numbers, divisibility rules, prime numbers, triangular numbers, square numbers, and so forth. These properties of study tend to be of a practical nature. Prime factorization, for example, is a huge area of research and has implications in cryptography. But before students get to cryptography, they are using these number properties to obtain least common multiples, finding common denominators, reducing fractions, completing the square, and then eventually progressing to some basics in combinatorics and number theory.

But what about the FUN numbers?!? We don’t have to mess around with “practical” number properties. Part of exploring mathematics is about just messing around. Now if you know me through this blog, you know I like palindromes.

Elba, I rave, ‘target nice C. Integrate variable.’

Numeric palindromes are fun and have some interesting properties. I gave a bunch of starters for anyone wishing to explore.

But if you are looking for something that’s not just about simply messing around, but rather would still help reinforce standard curriculum concepts, try this puzzle. You can modify it for where your students are mathematically. Here’s the puzzle:

Pick any range of a positive power of 10 — for example, 10 to 99 ($$[10^{1},10^{2}-1]$$), or 1000 to 9999 ($$[10^{3},10^{4}-1]$$). Now, are there more prime numbers in this range or more palindromic numbers? How do you know? How will you try to answer this?

Try this out! You will see that this can lead to a lot of good investigate math! Computing the number of palindromic numbers in a given range of a power of ten is a direct application of the Fundamental Theorem of Counting. And getting an estimate for the number of prime numbers in a range can be done in a number of ways, but probably the most famous / popular is the asymptotic formula $$f(x) = \frac{x}{\log(x)-1}$$. Here, $$f$$ is an approximation for the number of primes less than or equal to $$x$$.

If you want to start small, start with the range of integers in $$[10,99]$$. For students just learning about primes, it can be a little overwhelming, but once they’ve gotten familiar with these numbers, rather than have them memorize or otherwise repeat ad nauseum the primes, a little gophering task that has them compare the number of primes against the number of a different other type of number (palindromic, for example) can be a good way to reinforce the facts and concepts. Solving puzzles is what mathematics is about! 🙂

If you’d like to discuss further, feel free to reach out! Happy Mathing!

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