I started my magic
square by putting in my birthday:
8

27

19

92

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, rowwise, 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 squarea “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 strategyless, 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 middleside 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
whoknowshowmany different squares before he found one that worked for his
birthday without modernday technology, but also that he saw the value and
interest in making a magic square with these extra properties.
Fun post. Good sharing of method and I like the connections to Ramanujan.
ReplyDeleteI have a hint if you want it.
Look at Ramanujan's square and list out the entries.