Looking for a good math icebreaker for the new school year? Try working with palindromic numbers. There’s a lot of rich mathematics to explore without the need to construct contrived “real world” problems or having to worry about knowing where everybody’s math skill level is. There’s plenty for everyone. Let’s get started!

### Palindromic Numbers? What are they?

I swear as a kid I was taught that a palindrome was a *word* or *phrase* that could be read forward and backward (ignoring punctuation). Some common single word examples are: level, mom, dad, wow, pop, sis, noon, pullup, deified, kayak, repaper, madam, civic. Some common multi-word examples are: race car, demand named, edit peptide, evil olive, gateman’s nametag. Anyway, that’s neither here nor there.

I also swear that as a kid I was taught that there was a different name for palindromic *numbers*. But I can’t remember what it is. Maybe it wasn’t called anything else other than palindromic number. But what is a palindromic number? Basically, it’s any number that reads the same forward or backward (ignoring punctuation and disallowing leading zeros). Examples are 5, 11, 99, 343, 909, 1221, 4532354, 854458.

So what math can you do with palindromic numbers? Well, there’s a bunch! For the remainder of this article, if I say “number”, I mean positive integer. Also, I’m trying to give examples that will work across the spectrum of ability and knowledge. Feel free to modify accordingly for the grade you’re teaching and for your students’ abilities.

### Basic Starters

You’re the teacher. Pick any number of your choosing. Let’s say, 7346.

Find the first number greater than 7346 that’s a palindrome.

Answer: 7447

This is a simple and wonderful question, especially for your elementary students. It gets them to start looking at the number as a whole and reinforces some ideas about place value. I encourage you to try even- and odd-digit numbers. The structures are a little different!

Next question:

Find the closest number to 7346 that is a palindrome.

Your students may wonder why this question is any different from the first. And that’s a lesson in and of itself! A mild bias in how subtraction is taught is that we don’t emphasize enough the notion of absolute difference. This is a central idea when we talk about “closest”. The *closest* palindrome to 7346 is 7337 even though the *first* one *after* 7346 is 7447.

With these two questions we can start some more probing.

Next question:

Write all the palindromes between 7000 and 7999. What do you notice?

Answer: 7007, 7117, 7227, 7337, 7447, 7557, 7667, 7777, 7887, 7997.

See how quickly your students pick up on the pattern. (They might not get 7777!) Extend this for a five-digit palindrome. What’s even better is that for 1st and 2nd graders who don’t yet get to work with “big” numbers because the traditional arithmetic skills aren’t quite developed, is that here with palindromic numbers they can just write long, huge numbers and play with them.

### Fill In The Blank

Try this

Make all palindromes that fit this pattern: 8 _ _ 3 _ _

There are only ten solutions! They are 803308, 813318, 823328, 833338, 843348, 853358, 863368, 873378, 883388, 893398.

### Division Starters

Consider all the 2-digit palindromes: 11, 22, 33, …, 99. Notice they are all divisible by 11! But when we get to 3-digit palindromes, we’re not always so lucky — 101 and 111 aren’t divisible by 11, but 121 is.

Find all 3-digit palindromes are divisible by 11.

Your call if you want to allow them to use a calculator or if you want them to work out the division (or multiplication (if they see it that way!)) or if you want to introduce (or remind) about the divisibility rule for 11.

Answer: 121, 242, 363, 484, 616, 737, 858, 979

Notice anything interesting between consecutive differences? Why didn’t any of the 5_5 palindromes show up?

Next question:

Are all 4-digit palindromes divisible by 11?

Answer: Yes! But why? Shoot me a tweet if you’re stuck answering why.

More generally

Are all even-digit palindromes divisible by 11?

This is a great question to get some math reasoning going.

### Counting And Probability

How many 3-digit palindromes are there? 4-digit? n-digit?

Answer: Notice for 3-digit palindromes, there are 90 of them. There are also 90, 4-digit palindromes! There are \(9\times 10\times 10 = 900\), 5-digit palindromes. And there are 900, 6-digit palindromes! Can we generalize a formula?

What’s the probability that I will draw a 4-digit palindrome if I chose a 4-digit number at random?

This is a trickier question than it seems. Students get stumped trying to figure out how many 4-digit integers there are. So that’s a first exercise. Once they can figure out how many 4-digit integers there are, then finding the probability isn’t too bad. It’s just \(\frac{90}{x}\) where \(x\) is the number of 4-digit integers.

### Back To Arithmetic

Try adding two palindromes together. Do you get a palindrome? How can you guarantee that the sum of two palindromes will always be a palindrome?

This is a great question to get students experimenting. And a sneaky way to get them to add! What’s also nice is that students make up their own addition problems.

### Super Tricky Probability Game

Player A draws integers at random (uniformly) from \([100,999]\). Player B draws integers at random (uniformly) from \([1000,9999]\). Both players have a starting sum of zero. Let \(S_{A}\) be player A’s sum and \(S_{B}\) be player B’s sum. Every time player A draws a palindrome, she adds it to her current sum. Every time player B draws a palindrome, he add it to his current sum. Let \(n_{A}\) and \(n_{B}\) represent the number of draws for players A and B, respectively. The player with the fewest draws so that their sum exceeds 10000 is considered the winner. What is the probability that A wins?

This is probably best for a more advanced group to work out the problem exactly. But if you are teaching a programming class, this is not a bad exercise. For younger students, this can be a game to play. However, I’d recommend playing this out on the computer. If you are handy with a spreadsheet program, you can make this game directly in MS Excel, for example. If you really want, I can make a small web app that does this as well. Just shoot me a tweet. Plus, then I’ll know you got this far. ðŸ˜€

I also posted a small thread on this earlier that contains a few other starters. Check it out here (typo and all).

Try this as a first day #math activity (modify acvordingly for grade). Pick an integer, say, 7346. Ask students to find the first integer greater than 7346 that is a palindrome. #backtoschool

— M Shah (@shahlock) August 26, 2018

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