Doing Math: Magic Birthday Square

After class time on April 8th, I found one particular problem of Ramanujan both interesting and completely unfair.  I know all of the people in class that day felt the same way.  We were going through many of his accomplishments when it was shortly brought up that Ramanujan had a magic square that he had made out of his birth-date.  It made me wonder... Can any birth-date be made into a perfect square?  So of course I gave it my own go.  Luckily with today's technology, doing this is much easier than it had to have been in the past.  With the simple power of Microsoft Excel's "SUM" tool I could easily know if my perfect square would work.

I first started with my birthday, October 23, 1991 (10-23-1991), which adds up to be 143 (10+23+19+91).  So far the table is pretty empty, but it looks like the following,

10	23	19	91
-	-	-	-
-	-	-	-
-	-	-	-

I was trying to figure out what to do next.  I began with a simple guess.  I knew that all the squares (including the top left 2-by-2 square) had to be 143, so I took 19, added 1 to it and took 91 and subtracted one, leaving me with 90 and 20.  Now came the time for a decision, where to place the numbers.  I looked at the numbers and thought to try putting the larger number with the smaller number and vice verse.

10	23	19	91
90	20	-	-
-	-	-	-
-	-	-	-

So now I knew that the top left corner square was 143.  So again I went with what I thought to be a good educated guess.  I looked at the top right corner now.  I knew that the numbers need to add to be 33 so I first tried to do 32 and 1, with the same strategy as before, I thought to put the bigger number underneath the smaller number from the first row, so my magic square then looked like the following,

10	23	19	91
90	20	32	1
-	-	-	-
-	-	-	-

I was beginning to feel like I was getting somewhere.  I now had to take one more guess and the rest of the numbers should fall into place if they were to be right in this configuration, otherwise, I would have to start again.  I knew that the first vertical line was so far at 100 and I needed to get to 143, I also knew that the second vertical line was at 43 and needed to get to 143.  I took a wild guess and said why not try 2 and 98 for the two numbers for the second vertical line, I chose that it would be 2 then 98 in the vertical sense because I knew from the top that 23+19 was only 42, so it needed a lot more to get to be 143, so I was at,

10	23	19	91
90	20	32	1
-	2	-	-
-	98	-	-

I now knew 3 of the 4 squares from the square that makes up the middle two top numbers and middle two bottom numbers.  I did some simply subtraction (143-(23+19+98)) and found that the last number needed to be 3 in order to complete that square. I filled that one in next.

10	23	19	91
90	20	32	1
-	2	-	-
-	98	3	-

Since the vertical was now almost complete, it was only crucial to fill in the number above 3.  So I found what 143-32-19-3 is 89.  This now completes the third vertical row.

10	23	19	91
90	20	32	1
-	2	89	-
-	98	3	-

Next I will figure out the middle right square.  Since there is already a 32, 1, and 89, then 143 subtracted by those leaves 21.  So that gets filled in there and with that entry, there is the bottom right most square that can be filled in, subtracting 143 by 89, 21, and 3 leaves 30.  So we will fill those in below.

10	23	19	91
90	20	32	1
-	2	89	21
-	98	3	30

Using the same method, the two squares on the left can be filled in.  The middle left most square is then filled in with 31 since 143-(90+20+2) and then with that filled in the bottom left most square is filled in with 12 since 12=143-(31+2+98).  So the final square turns out to be as follows

10	23	19	91
90	20	32	1
31	2	89	21
12	98	3	30

After being elated that I found a magic square for my own birthday, I began to check myself.  I realized at that moment that my magic square was not completely correct.  I looked at the diagonals and the bottom and top middle squares and realized that these 4 sets did not add to the magic 143 number.  I tried for quite some more time, but did it to no avail.  I then questioned the thought that my birthday had a magic square.  I tried more numbers and worked through the problem the same way.  The first guesses turned out to be the closest I could get out of my 30 tries.  I then ended my trails at a magic birthday square believing that my birthday (along with the possibilities of many others) did not have a magic square associated with it.