Sunday, November 1, 2015

Euler's Square

Nature of Mathematics

This week we learned about the incredible Euler.  One of the activities we could do was to make an Euler Square of 16 playing cards. Elaine Young describes it in a MAA post, "In 1694, Jacques Ozanam posed the problem of arranging 16 playing cards in a 4 x 4 array such that no row or column contained more than one card of each suit and each rank. The solution forms an Euler square."  It's an Euler square because it combines two different Latin rectangles.  A Latin rectangle is like Sudoku, no number (or rank and suit in this case) is in the same row or column.  

Below are my two solutions to Euler's Square.  The first one is interesting because it has symmetry along the diagonals with the ranks of the cards.  It was really hard to get started on this one.  I started off by trying to make diagonals with one kind of number.  That didn't really work except for the Aces on the big diagonal.  From there I just tried to randomly move around cards to get something started.  The breakthrough was when I not only had the Aces on the diagonal, but also the Jacks on the other diagonal.  The second picture shows that each row, column, diagonal, and 2x2 square satisfies the requirements of no repeating rank or suit--including the middle 2x2 square.  I struggled with this one a bit too at first.  I knew I had to change up the diagonals so I attempted that first.  Then I tried to place each group of numbers as in a sudoku, with one number in each row or column and just filled in the gaps with the other numbers.  This worked a little, but then I had to modify and switch some based on suit.  I found that moving one usually results in changing three more.  But looking at it like a sudoku and filling in the blanks was helpful.

What I think is more interesting than just forming these Euler squares is thinking about if doing this is really math.  On one hand it's hardly more than arranging the cards in a puzzle type of way.  I certainly wasn't using any mathematical formula to figure out the arrangement.  However, there are many patterns to be seen in these squares and I believe finding patterns is a large part of mathematical principles.  For instance, if these squares had missing cards, you should be able to complete them based on the patterns within them.  Incidentally my Honors thesis is on rook polynomials which have similar properties to Latin rectangles, so I know there are ways to calculate the number of combinations of the cards.  There is a whole section of math dealing with combinatorics, which is counting combinations of elements.  So these squares can use math to describe and generalize them.  

Next question, if I think these are a part of math, was I doing math while I was making them?  This is a tough question.  I admitted I didn't use any of those formulas that count how many arrangements can be made.  But perhaps recognizing the patterns and following those pattern rules to fill out my square was doing math.  

It's hard to nail down exactly what is math and what is considered "doing math."  Instead of determining if this exercise was math, one could consider if this exercise is math, then what is math in general.  In that case I would definitely consider strategic pattern finding as a major part of math, in addition to the ability to numerically find conclusions that would be hard to find without math.  

Since I've decided to consider this math, what similar things could possibly be considered math?  Sudoku?  Jigsaw puzzles?  What examples can you think of?