The Dr Square Puzzle

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.



Comments 15

    1. Post

      Cheers, Dan! That confirms for me that there aren’t any loose ends hanging off my solution. I’ve got a proof for numbers above 1,000,000.

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  3. Lou Talman

    This reminds me of the Kaprekar constant: 6174.

    Begin with any four-digit number (zeroes count: 0073 is a four-digit number), N, that contains at least two distinct digits. Sort the digits of N into increasing order to obtain a four-digit number P. Sort the digits of N into decreasing order to obtain a four-digit number Q. Now subtract P from Q to obtain another four-digit number M

    Iterate, using the number M in place of N.

    Lou Talman
    Department of Mathematical and Computer Sciences
    Metropolitan State College of Denver

    I’ll let the reader figure out what this has to do with the Kaprekar constant.

  4. Andrew

    Awesome sequence! I lost a good chunk of the morning to trying to formalize my intuition that divisibility by three survives this version of digital root.

    Am I right in reading your puzzle that you only perform the digit sum once per cycle?

    Walking through the proof process with 8th graders was interesting. They were willing to deal with my “strange mod thingey” for discussing divisibility, less sure about referring to the digits of a number as A B C D … and then really outraged when after all that work, we decided to assume that our hard fought deduction WASN’T true. “But it’s gotta be true, Mr. C!”

    I don’t get to do many “car-crashes” with middle school kids, but when they show up, it’s awesome.

    =><= FTW

    1. Post

      You are right in your reading. Digital sum only happens once per cycle. I loved reading your post on taking the problem into a class. Glad you enjoyed the problem.

      What was the deduction you fought so hard for that turned out to be untrue?

  5. Sue VanHattum

    Can you tell us more about the student who created this puzzle? I’m going to use it in my math salon today, and I’d like to describe him. (Age and first name would be nice. Is he especially into math, or did he come to love it through working with you?) Thanks, Dan!

    1. Post
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