It can be fun to drive a problem all the way into the ground. I don’t always get to do that on paying projects, however sometimes I can do it wi

Calculating at Pencil and Paper Scale – Win Vector LLC

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2024-11-07 05:00:05

It can be fun to drive a problem all the way into the ground. I don’t always get to do that on paying projects, however sometimes I can do it with hobby projects. In this case I am going to re-solve Dudeney’s Remainder Problem again and again to argue that it can be solved at a pencil and paper scale. The problem is as follows.

A good idea for a solution is noticing that the answer is the greatest common divisor (GCD) of two differences of the puzzle inputs, such as 723217 - 480608 = 242609 and 508811 - 480608 = 28203. However, “every good idea becomes a lifetime’s work for someone” (Shriekback, Sacred City, “Ever force evolves a form”, 1992). Can we perform the remaining calculation explicitly at pencil and paper scale? Or must we use a canned algorithm on a computer?

This note is a somewhat involved series of demonstrations to show one can solve this problem “by hand” (or with minimal tools).