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Saturday, April 9, 2011

Ramanujan's magic square formula - Revisited

Ramanujan's genius (r) was discovered by Hardy (l)
At a very young age Ramanujan designed the following formula for a 3 by 3 magic square:

C+Q | A+P | B+R
A+R | B+Q | C+P
B+P | C+R | A+Q

where A,B,C are integers in arithmetic progression and so are P,Q,R.

Rewriting Ramanujan's scheme somewhat to
2Q+R | 2P+2R | P+Q
2P | P+Q+R | 2Q+2R
P+Q+2R | 2Q | 2P+R

where P,Q,R are in the Rationals, it is clear that every (P,Q,R) yields a magic square with constant number 3 (P + Q + R).

I conjecture that for any 3 by 3 magic square a triple (P,Q,R) can be found in the Rationals such that they fit the above scheme. Finding a proof for this is one of my 'problems'. Naturally, I would be very interested in any counter-example.


  1. I have studied one sample of ramanujan's magic square via microsoft excel:
    28 1 31
    23 20 17
    9 39 12
    the result is different from it. Then I started to discover properties from my own work. When I review ramanujan's sample, the properties I formulated applies to ramanujan's sample. So I further study and discover other things on magic square. Then I created formula's that follows the ramanujan's sample. Both my own work and ramanujan's sample were modified using an alternative variable z and add a "z-expander" to ramanujan's sample. The excel will view different number combination that follows definite proportions by changing z, central number and z-expander. As the central number increases, the sum increases. As z-expander increases, the difference between lowest and highest number increases, and as z increases, the difference between two consecutive number increases. Other interesting manipulation is by geometric rotation of the plane either by plane 90 degree rotation or by vertical, horizontal, and diagonal flipping. There are 8 arrangements that can be form from the 9 numbers of the 3x3 magic square. Using the microsoft excel gives a family of magic squares and of course in two forms (my own and ramanujan's).

    1. Please give details about your procedure.
      -Madhav Bapat.(

  2. Pls give a detailed information about theory..

    Thanks for the Information
    Education Portal


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