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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Dr. Square