Thursday, April 17, 2014

Doing Math: Wrestling with the Super Magic Square

I started my magic square by putting in my birthday:
I then tried entering values into the next row by just adding or subtracting a few numbers from the initial date. It seemed to work, row-wise, so I tried the next few rows. I realized this strategy works for getting the proper sum in each row, but it will leave me with too many low numbers on the left column, and too many high numbers on the right. So I tried accounting for that by starting with just my initial row, and working on one quadrant at a time (top/bottom left 4 cells, top/bottom right 4 cells). I went from the top left to the bottom right, adjusting previous quadrants to make the columns and rows turn out as necessary. This strategy took a while, but it proved victorious: I ended up with a square that had all of the rows, columns, and top and bottom quadrants each adding up to 146. However, in order for it to be a Ramunujan square--a “super” magic square—I had to figure out how to get the center 4 numbers, the diagonals, and the side 4 squares to add up to 146 as well.

I tried messing around with the numbers for a while, without much concrete success. As I was manipulating the numbers, virtually strategy-less, I realized that since the top row has to stay as it is, I already know how some of the other combinations of rows/columns will have to add up. For example, the two center cells of the second row must add up to 100, since the top center 4 cells must add to 146, and 27+19=46. I extended this to other cell values, and found that some various “required” values for combinations of cells are:

Using this new information, I tried making the middle-side 4 cells equal to 146 as well. After a a while spent fiddling, this was successful:

Feeling victorious, I decided to try and tackle the center 4 next. This was much harder than I expected, and in order to obtain the center 4, I sacrificed each side’s center 4.

After trying to wrestle with my magic square for another while longer, I decided that my magic square and I had come to an impasse. I decided that I would stop here, at a truce with my square: I let it retain the magic that it didn’t want to expose to me, and it gave slight concessions by almost conforming.

            Patterns and puzzles are what first drew me to math, so making a “super” magic square was very appealing to me. I enjoy knowing that these mind games are indeed math—while making this magic square I used basic algebraic skills, but I also found myself making linear combinations and trying to intuitively distribute the numbers amongst the cells. Even with the technology that makes adding instantaneous, it still took me a long time to get to the partial “super” magic squares that I did. Struggling to make my a magic square that has all of Ramanugan’s square’s properties really put Ramanujan’s genius in perspective. His genius wasn’t just that he was able to try who-knows-how-many different squares before he found one that worked for his birthday without modern-day technology, but also that he saw the value and interest in making a magic square with these extra properties.