Palindromic Addition

Posted by Greg MacDonald on

 

What can you do if children really would benefit from some addition practice, but they aren’t particularly interested in practicing addition?  

Of the many approaches that we might choose to take, I want to focus on one in this blog, and it’s based on one of the many practical pieces of advice that are spread throughout the works of Maria Montessori.  Here is what she said:

The teacher needs to be seductive, and can use any device - except of course the stick - to win the children's attention.
                    Maria Montessori: Education for a New World

Palindromic Addition is a “device” that can be introduced to the children (as Montessori suggests).  I have found that it often leads to a sudden surge in addition practice!

The word palindrome is derived from the Greek word palindromos, which meant “a recurrence”, or “a running back”.  Palindromes are words, phrases, sentences, numbers, and other sequences of characters (there are examples of music notation that are palindromic, for example) that read the same backward as forward.  

In English, we have palindromic words such as racecar and kayak. 

Was it a cat I saw? is an English palindromic sentence. (Capital letters, spaces between words, and punctuation are not included when assessing a palindromic sentence.)  If you were to accuse me having made a mistake here, I may respond palindromically: “I did, did I?

The Ancient Greeks created palindromes, which are only apparent when using the Greek alphabet.  There are palindromes in Asian and Middle Eastern languages, which are also not palindromic when translated into English.  The Ancient Romans enjoyed palindromes. One, apparently written to describe the movement of moths towards a flame, was:

In girum imus nocte et consumimur igni.
(“We enter the circle after dark and are consumed.”)

Clearly there are many Cosmic connections that we (and the children) can follow up when it comes to the idea of palindromes, but let’s return to the concept of Palindromic Addition. 

To conduct a Palindromic Addition, and so arrive at a Palindromic Sum: 

  • Start with any positive integer, reverse it, and add the two numbers.
  • Repeat the process with the sum obtained (so the sum, and the reverse of the sum, added together).
  • Continue to add each sum to its reverse, until the sum that you have obtained is palindromic. 

87, for example, generates a palindromic sum after four additions:

                                        87
                                    + 78
                                      165
                                   +561
                                      726
                                  + 627
                                    1353
                                 +3531
                                  4884 (A palindromic sum)

Mathematicians have discovered that, with the exception of 249 numbers, every number between 1 and 10,000 will generate a palindromic sum in under 24 additions.  The palindromic number 16,668,488,486,661 (for example) is produced after 20 additions by using the above approach with 6,999 (or its reverse, 9,996).

This makes me wonder if I can identify any of the 249 numbers that will not produce a palindromic sum.  My additions will have to be accurate.  One little error and there will be no palindromic sum! (Maybe if I work with some friends, and we all add separately, then check our sums each time, I can avoid that problem ...)

And I also wonder how many additions it will take for numbers that are special to me to generate a palindromic sum. Let’s see, my Mom is 34 and my Dad is 37 ... What if I put those ages together to make 3437, and start palindromically adding?

If you’re lucky enough to have Dean, Diana and Ed work on palindromic sums in your class, then you can probably find some way to use the following (palindromically):

Did Dean aid Diana?  Ed did.

 


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