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Luigi from Brooklyn writes, "My brother told me that 111,111,111 multiplied by itself is 12,345,678,987,654,321. Is that true or is that just another lie?"

Well, Luigi, it sounds to me like you and your brother have some trust issues! Does he often lie to you about math problems?

I can imagine you might have tried doing the multiplication on a calculator, and found that the calculator didn't have enough place values to display the result. Probably you got something ugly in scientific notation. There are calculators out there that will display enough digits to give you the answer. For example, the calculator on my computer will show the complete answer:

Hopefully that settles the dispute to your satisfaction. Of course, you don't need a calculator to do this; you can multiply it out by hand quite readily:

        111111111
      x 111111111 
        ---------
        111111111
       111111111
      111111111
     111111111
    111111111
   111111111
  111111111
 111111111
111111111
-----------------
12345678987654321

By the way, the title of this blog post is "Palindromic Multiplication," and there's a reason for that. A palindrome is a number (or word) that is the same both forward and backward. For example, "racecar" is a palindrome because, if you spell it backwards, you still have "racecar." This is an example where we squared a palindrome, and the result is another palindrome.

This doesn't always happen; it happens in this case because the multiplication doesn't result in any carrying (you can see from my multiplication above that the most we ever have added together in one column is 9 ones; if we had more than 9 ones, we'd have to do some carrying, which might mess up the palindromic nature of the result. So if we keep the digits small (mostly ones, zeroes, and maybe some twos), we may be able to find other palindromic results.

For example, multiply 121 by itself:

  121
x 121
  ---
  121
 242
121
-----
14641

But watch what happens if we try 131:

  131
x 131
  ---
  131
 393
131
-----
17161

This one is not a palindrome. Do you see why? In the third addition column we have 1 + 9 + 1, which results in a carry, and that throws off the symmetry of the result.

I know this is a lot more than you asked for in your question, but I think palindromes are very interesting, so I couldn't resist telling you a bit about them.

And finally, I leave you with the words of Napoleon, after he was exiled to the island of Elba: "Able was I ere I saw Elba." Now spell that backwards, and see what you get. :)

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