What Is A Palindrome? Words, Numbers And Dates
The short version
- A palindrome is a word, phrase, number or date that reads the same forward and backward. Racecar, level, 121 and 02/02/2020 all qualify.
- To test a number, write its digits in reverse order and compare. 4,554 reversed is still 4,554, so it’s a palindrome. 1,231 reversed is 1,321, so it isn’t.
- There are 9 two digit palindromes, 90 three digit ones and 90 four digit ones. Every four digit palindrome divides evenly by 11.
- Reverse and add is the classic puzzle: take 59, add 95 to get 154, keep going, and after three rounds you land on 1,111.
- Spaces, capitals and punctuation get ignored in word palindromes, which is how “A man, a plan, a canal: Panama” counts.
A palindrome is anything that reads the same in both directions. Take the letters or digits, flip the order, and if you get back exactly what you started with, it’s a palindrome. “Noon” is one. So is 838. So is the date 12/02/2021.
The word comes from Greek roots meaning “running back again.” Most people meet palindromes first as a word game, then again in math class, where they turn into a useful excuse to practice place value, reversing digits and looking for patterns. Below are the word examples, the number rules, the reverse and add puzzle, palindrome dates, and practice with answers.
What makes something a palindrome
One test, two steps. Write the thing out. Write it again backward. If the two versions match character for character, it’s a palindrome.
k a y a k
backward: k a y a k
They match, so kayak is a palindrome.
For words, the usual convention is to drop spaces, capital letters and punctuation before you check. That’s why “Madam, I’m Adam” counts: strip it down to madamimadam and it reverses onto itself. For numbers there’s nothing to strip, so you compare digits straight across and commas get ignored.
Notice what a palindrome is not. It isn’t the same as a symmetrical shape, and it isn’t a number that happens to have repeated digits. 1,221 is a palindrome; 1,212 is not, even though it uses the same four digits.
Palindrome words and phrases
Short ones first, since these are the ones kids can find on their own: mom, dad, eye, wow, pop, did, noon, level, radar, civic, kayak, rotor, refer, madam, racecar, stats, tenet.
Longer single words get rare fast. In most dictionaries the long ones top out around seven to nine letters, with racecar, rotator and redivider among the usual answers.
Phrases give you more room, because you’re allowed to ignore the spaces:
- Never odd or even
- Was it a car or a cat I saw?
- A man, a plan, a canal: Panama
- Step on no pets
- Taco cat
Names work too. Hannah, Bob, Anna, Eve and Otto all reverse onto themselves, which is why this topic lands well with a class of second graders holding name tags.
Palindrome numbers
A palindrome number reads the same right to left as left to right. Every single digit number is one by default, since there’s nothing to reverse.
| Digits | Examples | How many there are |
|---|---|---|
| 1 | 0, 1, 2, … 9 | 10 |
| 2 | 11, 22, 33, … 99 | 9 |
| 3 | 101, 111, 121, … 999 | 90 |
| 4 | 1001, 1221, 6776, 9999 | 90 |
| 5 | 10001, 12321, 45654 | 900 |
The counts come from the structure. A three digit palindrome looks like aba, so the first digit has 9 choices (it can’t be 0) and the middle has 10, giving 9 × 10 = 90. A four digit one looks like abba, and the same two free digits give 90 again.
How to check a number
Write the digits out with space between them, then read them backward and compare position by position. The place value habit matters here: a kid who reverses 1,050 has to produce 0,501, which is really 501, and that’s where the idea of a leading zero comes up.
Is 7,337 a palindrome?
Digits: 7 3 3 7
Reversed: 7 3 3 7
Match, so yes.
Is 4,040 a palindrome?
Digits: 4 0 4 0
Reversed: 0 4 0 4
No match, so no.
A pattern worth showing a kid
Every four digit palindrome is a multiple of 11. Try it: 7,337 ÷ 11 = 667, and 9,229 ÷ 11 = 839. It works because abba equals 1001a + 110b, and both 1001 and 110 divide by 11.
There’s a prettier one with repunits. 11 × 11 = 121. 111 × 111 = 12,321. 1,111 × 1,111 = 1,234,321. The digits count up to the number of ones, then count back down.
The reverse and add puzzle
This is the best palindrome activity for practicing addition, because the kid wants the answer. Pick any number. Reverse its digits. Add the two together. If the result is a palindrome, stop. If not, do it again with the new number.
Start with 59.
59 + 95 = 154
154 + 451 = 605
605 + 506 = 1111
Three rounds and you hit a palindrome.
Most small numbers finish in one or two rounds. 68 + 86 = 154, so 68 takes the same three rounds as 59. But 89 takes 24 rounds and ends up at 8,813,200,023,188, which is a great thing to hand a kid with a calculator and some patience.
Then there’s 196. Mathematicians have run it for hundreds of millions of digits without ever producing a palindrome, and nobody has proved it never will. A ten year old can understand the question and nobody on Earth knows the answer, which is a rare and useful thing to be able to say in a math lesson.
Palindrome dates
Write a date as eight digits with no slashes and check it like any other number. In the US format mm/dd/yyyy, 02/02/2020 becomes 02022020, which reverses onto itself. That one was unusual because it works in the day first format too, so it was a palindrome worldwide.
A few more eight digit palindrome dates in mm/dd/yyyy: 12/02/2021 gives 12022021, and 03/02/2030 gives 03022030. Some years offer several and some offer none at all, which makes hunting for them a decent homework break.
Shorter formats give you more. Written m/d/yy, dates like 1/2/21 or 5/1/15 read the same backward, so kids can usually find one within a few weeks of any given day.
Common mistakes
Checking only the outer digits. A kid sees 1,441 and 1,481, notices both start and end with 1, and calls both palindromes. Every position has to match its partner.
Forgetting that a reversed number can lose a digit. Reverse 1,050 and you get 0501, which is just 501. That trips up the reverse and add puzzle when a number ends in 0.
Confusing palindromes with repeated digits. 5,555 is a palindrome, but so is 5,225, and 5,525 is not. Repetition isn’t the test; mirroring is.
Stopping the reverse and add puzzle too early. If the sum isn’t a palindrome, the round isn’t over. Kids often stop at a number that merely looks tidy, like 605.
Applying word rules to numbers. Spaces and punctuation get ignored in words. Digits never do.
Practice problems
- Is 4,554 a palindrome?
- Is 1,231 a palindrome?
- Which of these are palindromes: level, lemon, kayak, kayaking?
- List every two digit palindrome number.
- What is the next palindrome number after 2,992?
- Run reverse and add on 59. How many rounds does it take, and what do you land on?
- How many three digit palindromes are there in total?
- Divide 7,337 by 11 to check the rule that every four digit palindrome is a multiple of 11.
Answers
- Yes. Reversed it’s still 4,554.
- No. Reversed it’s 1,321.
- Level and kayak. Lemon reverses to nomel, and kayaking reverses to gnikayak.
- 11, 22, 33, 44, 55, 66, 77, 88, 99. That’s nine of them.
- 3,003.
- Three rounds: 59 + 95 = 154, then 154 + 451 = 605, then 605 + 506 = 1,111.
- 90. The first digit has 9 options and the middle digit has 10, so 9 × 10 = 90.
- 7,337 ÷ 11 = 667, with nothing left over.
If your child can’t do these
Palindrome problems are a sneaky diagnostic, because they hide a place value test inside a game. A kid who can’t reverse 1,050 correctly, or who writes the reversal of 308 as 83 without noticing what happened, is telling you something about place value rather than about palindromes.
The reverse and add puzzle catches a second thing: multi digit addition with carrying. Watch which round they start making errors in. If 605 + 506 goes fine but a four digit sum falls apart, the carrying is the gap, and no amount of palindrome practice will fix it.
The fix is to go back a level and rebuild, which is faster than it sounds. Our guide to math skills by grade level shows what should be solid by the end of each year, and Is My Child Behind In Math? lists the signs worth acting on. Puzzles like this one also earn their keep with kids who dread worksheets, which is half the battle when math anxiety is in play.
Frequently asked questions
What is a palindrome in simple words?
Something that reads the same forward and backward. Flip the letters or digits around and you get the same thing you started with. Mom, radar, 121 and 9,779 are all palindromes.
What is a palindrome number?
A number whose digits are in the same order read either way, like 66, 303 or 45,654. Every one digit number counts automatically.
Is 0 a palindrome?
Yes. It’s a single digit, so reversing it changes nothing. The same goes for every number from 0 to 9.
What is the longest palindrome word in English?
Ordinary dictionary words run out around seven to nine letters, with racecar, rotator and redivider near the top. Longer ones exist but are mostly technical terms or coinages. Phrases can run as long as you like, since you’re allowed to ignore the spaces.
What grade do kids learn about palindromes?
Word palindromes show up as early as first or second grade. Palindrome numbers usually appear in second to fourth grade alongside place value work, and the reverse and add puzzle fits nicely once multi digit addition is solid.
Find the actual gap
Games like this one are a good way to spot trouble, but spotting is only half of it. If reversing digits or carrying across columns is where your child comes unstuck, you want to know exactly which earlier skill is missing before you spend another evening on it. Math Anon begins with a short adaptive assessment that adjusts to each answer and finds those skills, then hands you a written report in plain language.
After that it’s about 20 minutes of daily practice pointed straight at the gaps that turned up, with a progress report each week. The assessment costs $49 and we credit every dollar of it back on your first month. Start the assessment or take a look at pricing first.
