Choose your own adventure!

When I was in middle and high school, I came across a series of books called "Choose Your Own Adventure."  These were really exciting and I read many of them.  I realized this past week, that doing research at the Library of Congress and doing math have a lot in common with these books (including the excitement I felt as a kid.)

While wandering through some LoC collections of online books about games, I came across Games of skill, and conjuring: including draughts, dominos, chess, morrice[1].  "Skill and CONJURING"?!  My inner 13-year-old was immediately intrigued and I spent a bunch of time paging through.  I found the games "Morrice", also known as "Nine-Men's-Morris", and the game called "Fox and Geese".  I had heard of these games before, but never really played them, and since I was looking for things I and other math teachers might use, I found online versions of the games and played against the computer a few times.  I lost.  Every time.  The rules sounded simple enough, but the strategy is something I will need to think about a lot more!  The fact that a computer was always picking the best moves tells me there is some sort of algorithm that can be used, so I do plan on using these games in my classes.  Even though I don't know the strategy.  I'll learn something along with my students.

I continued through the book (choosing a different adventure), and found a bunch of puzzles.  One that intrigued me was this one, from page 91:

Based on the third figure, I realized that the rectangle (I decided that figure 1 was not just a parallelogram, but also a rectangle.) had to be cut into some sort of stair-step shape, and looking at the solution a few pages later, my thinking was confirmed:

Using the stair-steps to get what looked like a square (figure 2) was more intriguing, because I wondered if any rectangle can be cut this way in order form a square, or was there something special about the rectangle in the original problem.  So I posed myself this problem:

Is it possible to cut any rectangle into two equal pieces, so that the pieces may be rearranged to form a square?  I gave myself the constraint of the pieces being equal, and decided to stick with the stair-step division, so that I could at least answer that much of the question.

I needed some way to represent this problem, so I quickly sketched some diagrams and possible algebraic representations on a legal pad:

I didn't know exactly what I was doing, and the algebra was not obvious right away.  There were lots of "adventure paths" here.  I had to decide how many stairs and what variables might represent which aspects of the picture.  I knew that the area of the rectangle and the area of the square had to be the same.  I quickly decided that all the stair-steps needed to be the same size in order to fit the pieces back together.  I also figured out, based on the diagram in the book and some diagrams I drew (the bottom left one in particular) that there would be some number of divisions along the short side of the rectangle and one more division along the long side.  (As I write this, I am wondering if that's always true, or could I arrange the steps another way ...)  I didn't get very far with the problem, and I had other things to do, so I set it aside.

That night, while I was getting ready for bed, my mind wandered back to the problem, and I thought about focusing on the ratio of the two sides of the rectangle. If I divided the short side, x, into n pieces and the long side, y, into n+1 pieces.  Representing the two sides of the square using those variables, I could probably come up with a ratio of the sides.  Since I was tired, I made a note about this idea and went to sleep.  The next morning, I played with this idea and quickly drew a diagram of the rectangle and the resulting square using these pieces, and did some algebra to determine the ratio of y to x:

Since the number of divisions has to be a whole number, I made a table to determine some of the possible rectangles.  I realized that the ratios did not look right, as y/x should have been bigger than one, rather than less than one, and the the pink and black diagram showing the 3x4 rectangle certainly was not going to form a square.  My algebra looked okay, but something was seriously wrong.  When I met an untimely end in a Choose Your Adventure Book, I always backed up a page or two and tried again, so ...

I rechecked my diagrams of the rectangle and resulting square, was happy with the variables, and got a new sheet of paper to start the algebra again.  I immediately saw that I had changed the n+1 from the diagram to n-1 in the first run at the algebra, so redid the algebra on a new sheet, and made a corresponding table of values as well as a few diagrams to verify the work:

It does not surprise me that the base area of these rectangles/squares were perfect square numbers, and I did not think it unreasonable that the table was skipping the odd square numbers; but why was it skipping some of the even square numbers as well?  The resulting side lengths of the squares were increasing in a quadratic sequence. Why not a side length of 8? or 16? Maybe the fact that these are powers of two has something to do with it.  But then why is a side length of 14 not on the list?  Is there something about the factor pairs of the square numbers that worked?  Maybe there's a different way to divide up a rectangle to get these sizes.

As I started to pursue these ideas, more questions and ideas popped into my head, and I realized the adventure could continue:

• The numbers in each ratio are consecutive perfect squares. What if they're not consecutive?  What happens when I try to make a square?  What happens if they're not perfect squares, like 3/1; how close to a square can I get?  Maybe it's time to get out the ruler and scissors to create some models.
• I can take any square, and split the sides however I want, making the square 5x5, 6x6, or 14x14.  The units don't matter; it's still a square.  Can I divide the squares into two congruent stair-step shapes to get a rectangle?
• The limit of the ratio y/x approaches 1 as the number of divisions increases.  What does this mean?
• Are there other ways to cut a rectangle in order to get pieces that rearrange to a square?  I limited myself at the beginning to two congruent pieces that had a stair-step shape.  What if I removed or eliminated one of these parameters?
• As I was copying the pictures for this post, I realized that the original problem said "parallelogram", not "rectangle".  Is it possible to cut a parallelogram that is not a rectangle into shapes that could form a square? (I know how to cut a parallelogram to form a rectangle, so this should be possible ...)
• And what about the original solution shown in the book?  That showed five steps in one direction and six in the other, so according to my formula, the ratio of sides has to be 36/25, right?  The solution also mentioned dividing "a piece of card"; a 5x7 card has a side ratio close to 36/25.  Maybe that's what the puzzle was using.  Did they use 5x7 cards in 1865, when the book was published?

Now I have lots more to explore, including the history of stationery.  I don't know if I'll answer all these questions, because working on one of them may lead to other more interesting paths.  Or some other completely different and shiny math problem will show up and beg me for attention.  I'm always interested to see where the next path takes me ...

References

1. Games of skill, and conjuring: including draughts, dominoes, chess, morrice. George Routledge and Sons, London, 1865.