Is a piece of the puzzle missing?

Introduction

Whenever I go home for a holiday break (home for me is Pittsburgh, Pennsylvania, by the way), my routine tends to change from the normal, and while I like for my breaks to be productive, some more time for relaxation is inevitable. One activity of choice for me and my family as of late has been solving jigsaw puzzles.

I had customized and ordered a Shutterfly puzzle of our family’s cats as a Mother’s Day present in 2020. We didn’t have time to put it together back in May, so we decided to open it up this weekend and give it a shot. We sorted out all of the “border” (edge and corner) pieces of the puzzle in order to put together the border first. In doing so, we realized that some border pieces were missing, even though all the border pieces that had been sorted seemed like they fit together. In this article I’m going to detail how I used some basic number theory to figure out how many border pieces were missing, and which portion of the border they belonged to (top, left, bottom, right).

Picture of our partially solved jigsaw puzzle, with only the outer frame in place

Solving the Puzzle

The puzzle, when fully solved, has 520 pieces. I made the assumption that the puzzle can be modeled as a rectangle of unit cells: that is, the completed puzzle looks like a large grid of tiny squares fit together. when viewed from afar. With this assumption, I figured there was certain number of…