Palindromes as Computational Practice

Purposes:

1. 1. To Promote interest in mathematics
2. 2. To provide practice in addition of whole numbers in a context other than pure drill

I. What is a palindrome?

• A palindrome is something, which when written backwards or forwards, is the same thing.
• A palindrome may be a word such as mom, pop, sis, toot, or radar.
• Some more ardent palindromists have have developed long phrases or sentences that are palindromes. Some examples are:

  1. Madam, I'm Adam.

  2. Palindromists looked diligently for an appropriate response, but for a long time the best they could muster was "Sir, I'm Iris."

  3. Finally, some palindromist came up with "Even am I, man, Eve." Sure, it's convoluted, but it is a palindrome.

  4. Do nine men interpret? Nine men, I nod.

  5. A man, a plan, a canal: Panama.

  6. One of the best is Napolean's lament when he was banished from France to the Isle of Elba: "Able was I ere I saw Elba."

II. Number Palindromes

Number palindromes, unlike sentence palindromes, are easy to make. All you have to do is just to write down a number palindrome. However, the thing that makes number palindromes useful is having them arise through computation. This is the way palindromes provide practice in computation in a context other than repetitive drill. Here are some ways to do this:

1. 1. List the sequence of numbers 11x11, 111x111, 1111x1111, and so on until a palindrome does not result. Before you start, can you predict at what point the result will not be a palindrome? Here are the first three computations. See if you can find the pattern that is emerging.

   a. 11x11 = 121

   b. 111x111 = 12321

   c. 1111x1111 = 1234321

   2. List the sequence of numbers 11, 11 squared, 11 cubed, 11 to the fourth power, and so on ubtil you do not get a palindrome. Here are the first three computations:

   a. 11

   b. 11x11 = 121

   c . 11x11x11 = 1331

   3. The system leading to palindromes that provides the most practice in addition of whole numbers is this:

   a. Write down any whole number,

   b. Reverse the order of the digits.

   c. Add the two numbers.

   d. If this produces a palindrome, you are through.

   e. If this does not produce a palindrome, reverse the digits on your sum and add the new number to the number you just reversed.

   Example:

   78 + 87 = 165

   165 + 561 = 726

   726 + 627 = 1353

   1353 + 3531 = 4884 (Which is a palindrome.)

A palindrome occurs much more quickly with some numbers, and takes much longer with others. A good computational exercise is to apply this process to all numbers from 80 through 100. Most of these choices are easy, but 89 is a scorcher. It finally becomes a 13 digit palindrome. Note that 98 is included in the above sequence, and a student should notice that 98 is the same as 89 for this process.

Note: Do not try 198. This number has been tried by a powerful computer and a palindrome has never been found. No one knows whether the process would eventually lead to a palindrome.