Dr. Square

Much like the Collatz conjecture, the Dr. Square Puzzle (from the blog mathforlove) is an intriguing problem.

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.

Naturally, I wrote a python program to check the first million number and so far it looks like they are right. If you take off the comment symbol (#) from lines 6, 20, 23, and 24 then it will print the path to each number.

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4 Responses to Dr. Square

  1. D.Guy says:

    Here are some simpler puzzles that I also enjoyed:

  2. Ben says:

    Are all numbers suppose to return to 1 or end up in an endless loop? One example that doesnt end up as one, but does get into a loop is 12. 12^2=144. 1+4+4= 9, 9^2=81, 8+1=9, 9^2=81… I am possibly reading this problem incorrectly.

    • Dan says:

      All numbers should(?) end up in a loop, but not all the loops are stuck at 1. I noticed that all the numbers up to 1 million end up in either a 1, 9, or 13 loop. Not sure for numbers past 1 million. 🙂

  3. Jack says:

    You need not try millions. For all numbers larger than a much smaller N squaring and summing the digits decreases them. I let you work out the N, and the proof by induction that you do get the 1, 9 or 13 loop.

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