A Magical Birthday Square (Doing Math

After learning about the great accomplishments Srinivasa Ramanujan did with such limited resources and at such a young age I was astonished. Then I found out he came up with a magic square with 22, 12, 18, and 89 in the first row, which is his birthday! I thought it was so cool that I had to try it for my own birthday. So my first row consisted of 1, 7, 19, and 92. That meant my target sum was 119. I wasn’t sure where to start at first and thought I was going to have to do a lot of guessing, but then I saw that when finding the target sum of the top right corner of the 2×2 I had 19 and 92 already. When you added those two up you get 111, which meant the two bottom squares of the 2×2 could either be 6 and 2 or 5 and 3. This made guessing a lot easier, so I went with 6 and 2, with the 6 under the 19 and 92 under 2. I did it in this specific order because I will need those rows to add up to 119 as well and since 92 is such a big number I wanted to place it with as many small numbers as possible. (If at any time you get confused on where I put any numbers, there is a picture of my square).

                After I made this guess the rest of the 2nd row was easy to figure out because I just had to look at the top middle 2×2 and the top left 2×2. So for my second row I had 24, 87, 6, and 2. This adds up to 119. I then looked at the diagonal that had a positive slope. It already had 92 and 6 so the sum was 98. This meant that my possible options were 3 and 18, 4 and 17, 5 and 16, 9 and 12, or 10 and 11. Not quite as nice of odds for guessing so this is when I looked at the four corners. I already had 1 and 92, which is a sum of 93. I knew that of the numbers in that could be in the diagonal had one of the same number as the four corners. So the possible numbers for the bottom to corners was 3 and 23, 4 and 22, 5 and 21, 8 and 18, 9 and 17, 10 and 16, 11 and 15, or 12 and 14. That unfortunately didn’t eliminate any so then I looked at the last column which had 92 and 2, a sum of 94. I was hoping this would eliminate some possible options for the four corners. So the possible options for the last column were 3 and 22, 4 and 21, 5 and 20, 8 and 17, 9 and 16, 10 and 15, 11 and 14, or 12 and 13. At this point I’m frustrated because I am not catching a break, so I go back to the diagonal I looked at originally and I took a guess. I choose 10 and 11 thinking that they were fairly neutral numbers (neutral meaning they won’t screw up the whole square by a lot if the numbers didn’t fit). Now that I had that I could find the middle 2×2 since I already had 87, 6, and 11. So I got 15. This allowed me to find the other diagonal since I had 3 out of the 4 squares and I got 16. I can find the first and last column since I had only one square blank in each so for the first column I had 84 and the last column I had 9. From there I was able to fill in the rest of the square.

                Once I filled all the squares in, I started to check my work. I did this while in my MTH 122 class and as I added up each row getting a total sum of 119 my heart starts to speed up. I then add up each column and again a total sum of 119 for each. My heart I start to get a rush of adrenaline and my leg starts shaking. I think to myself, “Could it be that my birthday, just like the great Ramanujan, makes a perfect square!” I think check my diagonals, again 119! I start to lose my breath. Then I check the middle 2×2 and the middle edges. All add up to 119. At this point I’m freaking out. I have to try and contain my excitement since I’m sitting in class. I then check the top middle 2×2 and bottom middle 2×2. Boom! 119 like a champ! I then take a picture and post it in our Facebook group. After I made the post, I start to double check my work… I see I didn’t check the left middle 2×2 and right middle 2×2. And they don’t look pretty. My numbers are way off. Instantly my dreams are shattered. All the excitement I had been feeling deflates at rapid speeds. I start to look at what I could change to fix the square so it would work and I realize that it is impossible. Since 1 is in the first column I need to have a big number somewhere in that column. Ideally, I would want to put it in the bottom square so I wouldn’t screw up the middle right 2×2, but because 92 is already in the top right corner I can’t do this (because 4 corners and the diagonal wouldn’t do this). This means that I have one really big number for the first column, which is fine I can accommodate for that. It is not ideal but it still can be done, but because the top right corner 2×2 under the 19 and 92 can only be 2 and 6 or 3 and 5 it means that the top middle have to be a large number. Thus you will always have two really big numbers putting you over 119. I unfortunately was unable to get my birthday work, but I came close and got to feel like I was with the elite, like Ramanujan, for at least 5 to 10 minutes. Maybe the birth of my kids will be in the magic square club.

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One thought on “A Magical Birthday Square (Doing Math”

  1. “While in your Math 122…”?

    Good doing math think aloud. I think where the trouble is is the take a guess stage… that’s a good way to start, but you need a reason as you revise. That will get you a regular magic square, but things are tight in Ramanujan’s.

    If you look at Ramanujan’s numbers you might notice more structure that will help you finish.

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