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Palindromic Words and Numbers


I was getting ready to pay for my meal at a buffet when I noticed the cashier's name tag.  It read "Asa" to which I replied, "Your name is a palindrome!"  The cashier just stared at me in disbelief.  I explained that a palindrome was letters that read the same backwards as forwards. Because you could read his name forwards and backwards, it qualified as a palindrome.  He replied that he remembered a math teacher talking about those because of patterns (I love that math teacher), and he remembered the phrase "race car" was a palindrome.  We then began sharing palindromes that we knew such as radar, level and madam while my family waited impatiently in line. (Sometimes they have little patience with my math conversations.)

The word palindrome is derived from the Greek word pal√≠ndromos, which means "running back again". A palindrome can be a word, phrase or sentence which reads the same in both directions such as: "Eva, can I stab bats in a cave?" or "Was it a car or a cat I saw?" or "Rats live on no evil star."

But did you know there are also palindromic numbers?  A palindromic number is a number whose digits are the same if read in both directions (as seen on your left).   Whereas "1234" is not a palindromic number, because backwards it is "4321" which is not the same. 

Suppose a person starts with the number one and lists the palindromic numbers in order: 11, 22, 33, 44, 55...etc.  Can you continue the list? 

Did you notice that palindromic numbers are symmetrical?  Look carefully at the 17371 shown above.  It is symmetrical (when a figure can be folded along a line so the two halves match perfectly) on either side of the three whether read left to right or vice versa.

Palindromic numbers are very simple to generate from other numbers with the help of addition.

Try this:

1. Write down any number that has more than one digit. I will use 47.
2. Write down that number in reverse beneath the first number. (See illustration below.)
3. Add the two numbers together. (121)
4. The sum of 121 is undeniably a palindrome.

Try an easy number first, such as 18.  At times you will need to use the first addition answer and repeat the process of reversing and adding. You will almost always get a palindrome answer within six steps.  Try one of these numbers 68 or 79.  Be careful because if you pick a number greater than 89, arriving at the palindromic answer will take more steps, but it will still work.  (See the two examples below.)

But don't try 196!  In fact, avoid it like the plague!   A computer has already gone through several thousand stages, and it still hasn't come up with a palindromic answer!

Example #1:

Start with 75.

Reverse 75 which makes 57.

Add 75 and 57 and you get 132.  The answer 132 is not a palindrome.

SO reverse 132, and it becomes 231.

Add 132 and 231, and the answer is 363. Since 363 is a palindrome, we are done!


Example #2:
Begin with 255.

Reverse 255 to get 552.

Add 255 and 552.  The answer is 807 which is not a palindrome.

SO reverse 807 to get 708.

Add 807 and 708.  The answer of 5151 is not a palindrome.


SO reverse 1515 to get 5151.

Add 1515 and 5151 which is 6666.  This is a palindrome; so, we are done!

4 comments:

Pamela Kranz said...

It's nice to know I'm not the only one that gets a kick out of palindromes. It happens frequently and I always happily announce, "Palindrome!!" And I usually have to explain it to the clerk. ;) Fun!
Pam

Les Vrolyk said...

Hi Scipi! Thanks for sharing this interesting info about palindromes. I was not aware that you could add numbers with this method to create number palindromes...you learn something new every day. But what is up with 196??!! - Les

TheElementary MathManiac said...

I love exploring this idea with kids!


Tara
The Math Maniac

Anonymous said...

This is awesome. I never knew that trick of adding to get palindromes.
Are you sure palindromic numbers are symmetrical?
If you folder 17371 the bottom and the top could not match up perfectly or if you folded it along a vertical line the 7's wouldn't match up and the left half of the 3 wouldn't match the right half of the 3.
I think a number like 10801 would be symmetrical depending on your font.