The Dr Square Puzzle

November 11, 2011

One of my students yesterday shared an idea with me that he had kicking around in his head for a while, which he called continued multiplication. Essentially, he used the steps in a multiplication process as an iterative process to give him a new multiplication problem, which he would loop again and again. We clarified some of the rules (too complex to mention in full detail here), and I sent him home to investigate it further.

He came back having tried a few examples, but without having found any major breakthroughs. The main problem was that the numbers tended to get big fast. Sometimes these puzzles, I suggested, needed a certain balance between getting large and shrinking. The best of them–like the Collatz Conjecture–strike this balance perfectly. We needed a way to shrink the size of our numbers.

With some playing around, we came up with what I think is an excellent (and solvable) puzzle. He dubbed it the Dr Square puzzle, because it involves one of the steps in taking the digital root (dr) and squaring numbers. Here’s how it goes.

Step 1: Choose a starting number.
Step 2: Square the number.
Step 3: Sum up the digits of that number.
Step 4: Repeat steps 2 and 3 until you understand what’s going on.

Example. Let’s take the number 26. Squaring it gives 676. The digital sum of 676 is 19. Squaring gives 361. Digital sum of 361 is 10. Squaring 10 gives 100. Digital sum of 100 is 1. Squaring gives 1. Digital sum gives 1. So we stay at 1 forever once we get there.

More briefly, we could write 26 –> 676 –> 19 –> 10 –> 100 –> 1 –> 1 –> 1 –> etc.

We called this the 1 loop.

We discovered three loops so far, which we’ve called the 1 loop, the 13-16 loop (13–>169–>16–>256–>13), and the 9 loop (9–>81–>9). While we conjecture that these are the only three loops, we don’t have a proof yet.

Here’s our data so far. See any patterns?


At this point, the questions are like dogs scratching at the door, beggin to be let out. So here they are:

  1. Are there just three loops? Or are there others that we haven’t discovered yet? How can you prove it?
  2. Is there a quick way to see which loop a number will end up in?
  3. Will every number end up in a loop? Is it possible that something else could happen?

I’m almost certain that there are nice, findable answers to all these questions (because I’m pretty sure I’ve almost found them). Ideas? Questions? Put them in the comments, and I’ll respond. I’ll come back tomorrow to give some hints, and I’ll get to a solution within the next few days, if I can.

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Touzel Hansuvadha
3 years ago

Whoa! So cool–I want to share this with my daughters and my students. Thanks for sharing this!