Quandaries and Queries
 

 

Hello my name Lillian.
 I am a student in the 10th grade.  My class is studying palindromes.  I was wondering why

12, 21, 13, and 31 are the only double-digit numbers whose squares are the palindromes of the squares of the palindromes of double-digit numbers.  I was wondering why it works for these numbers, and if only these numbers work this way.

Thanks

Lilly

 

 

Hi Lilly,

That's a really nice observation.

Lets look at 13 and 31.

  1 3            3 1
1 3   3 1
--------- -------- --------   -------- -------- --------
  3 9     3 1
1 3     9 3  
--------- -------- --------   -------- -------- --------
1 6 9   9 6 1

It's the beautiful symmetry that makes this work. A problem arises however if one of the digits is larger than 3. In this case the square of the digit is a two digit number so in the multiplication there is a carry that destroys the symmetry.

By the way, don't 11, 22 and 33 meet your conditions?

Chris and Penny 

 
 

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