## Mathematical Palindromes

A palindrome is a word or a number that can be read the same way in either direction. For example, MOM is still MOM if read backwards. “A man, a plan, a canal, Panama” is a famous palindrome about the building of the Panama Canal. Palindromes can also be numbers. 121 is still 121 if read from the other direction.

Time & Date says:

*A Palindrome Day happens when the day’s date can be read the same way backwards and forwards. The dates are similar to word palindromes in that they are symmetrical. Because date formats vary from country to country, not all dates that are be considered palindromic in one kind of date format are Palindrome Days in another. For instance, July 10, 2017 or 7-10-2017 is a palindromic date in the m-dd-yyyy format, but isn’t if you write the date as mm-dd-yyyy (07-10-2017); as dd-mm-yyyy (10-07-2017) or as yyyy-mm-dd (2017-07-10).*

So here in the USA this is the start of a palindromic week: 7102017 or 71017 through 71917!

Younger students will have fun investigating palindromic numbers. Students can discuss if they think a two-digit number can be a palindrome. Palindromes can also be extended to time and money. Be careful of the decimal point in money and the colon for time. 9:09 am is not truly a palindrome since it is not the same on each side of the colon. But 10:01 is a palindrome. The same idea applies to money. What about calendar dates and years? When will there be another year that is a palindrome? What was the last year that was a palindrome? What about dates, if written in numerical form? 11-02-2011 was a palindrome date. Pi day in 2013 was another palindrome: 3-14-13! Dates can be written in different forms to determine if they are palindromes.

An interesting investigation for mathematics students of any age is the creation of palindromic numbers. Many numbers can be made into palindromes by adding the number with its digits reversed.

Examples: 41 + 14 = 55 342 + 243 = 585

Students can discuss why these examples work and why other numbers require the process to be applied twice.

Example: 57 + 75 = 132 then 132 + 231 = 363

This can be extended to other number sets that require 3, 4, or more applications of the process to create a palindrome.

Students can use a 100 chart to begin their investigation. Have them color code the numbers on a hundreds chart by those that require one application of the reverse addition, two applications, three applications, four applications, five applications, and six applications. The numbers 89 and 98 require even more applications! Some students may benefit from using a calculator with this activity as it helps with the larger numbers.

Have students see if they can determine how many two-digit numbers are palindromes, how many three-digit, etc. Students can then extend the study of palindromes to multiplication to see if they can take two numbers and multiply to get a product that is a palindrome. Or, for secondary students, what perfect square numbers that are palindromes have palindromes for roots. For example, the palindromic perfect square 121 has the palindrome root of 11.

Danica McKellar is credited with this riddle:

“What did the mathematician say when she was offered cake?” |

“I prefer pi.” |

On an episode of Big Bang Theory, Sheldon wrongly makes a statement about a palindromic number. Have students watch the clip and see if they can find Sheldon’s mistake!

It is the episode where he gives his “perfect” number – 73. Big Bang: https://www.youtube.com/watch?v=TIYMmbHik08

Math Goodies website: http://www.mathgoodies.com/palindromes/ has a section on palindromes that has some interesting palindrome puzzles.

NRich has some resources on Palindromes to check out:

Palindromes article:

*I was in Bewdley, in Worcestershire recently. It’s a very old town and well known for a pretty bridge over the river in the centre of town. But it’s also known for another reason, one that most people don’t know about. Mr Edward Benbow, from Bewdley, once held the palindromic record! “The what?” you may be asking….*

**Some palindrome books include:**

*Sit on a Potato Pan Otis* &

*Go Hang a Salami! I’m a Lasagna Hog!*

both by Jon Agee

*Too Hot to Hoot *by Marvin Terban

**Have fun this week and enjoy palindromes!**