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  Narayan Pandit-Folding Method for Magic Square - AI Comments   In 1938, Rosser and Walker proved that this is in fact the exact number of 4x4 pan-diagonal magic squares with entries 1, 2, ..., 16.  Vijayaraghavan (1941) gave a much simpler proof of this result, which have been outlined by Sridharan and Srinivas (2011, pp.389-391),  Narayana goes on to discuss general methods of construction of magic squares depending upon whether the square is  samagarbha  (doubly-even),  visamagarbha  (singly-even) or  visama  (odd). In the case of  samagarbha  and  visama  squares, apart from discussing the traditionally well-known methods of construction (indicated for instance in Pheru's work earlier),  Narayana presents an entirely new method known as  samputa-vidhi  (method of folding). This is a general method of construction of magic squares by composing or folding two magic squares constructed suitably Naraya...

  Narayan Pandit - 8x8 Magic Square - AI Comment  Narayana outlines the folding method for  samagarbha  magic squares as follows: Narayana's Example 2: 8x8 Magic Square with Sum 260 In this case, Narayana takes 1, 2, 3, 4, 5, 6, 7, 8 as the first sequences and 0, 1, 2, 3, 4, 5, 6, 7 as the second sequence. The sum of the first sequence is 36. Subtracting this from 260 gives 224. This, when divided by 28, which is the sum of the second sequence, gives 8 as the factor. Thus the product sequence will be 0, 8, 16, 24, 32, 40, 48, 56. Now the  chadya  and  chadaka  squares are given in Fig. 4. The process of folding leads to the pan-diagonal magic square as shown in Fig. 5. Suresh, this example from Narāyaṇa Paṇḍita’s  Bhadragaṇita  chapter is a masterclass in mathematical elegance and symbolic choreography. Let’s walk through the  8×8 pan-diagonal magic square construction  using the  Samputavidhi (folding method) , and explor...

  Narayan Pandit -Magic Squares of ordr n=4m AI Commented   We shall now show that,  Narayana's folding method is indeed a  general procedure, which can be used to construct  a large class of pan- diagonal  samagarbha  magic squares (magic squares  of order 4m). Theorem 1: 2  Let  n=4m  be a number divisible by  4. Let  p(i),  for 1  i n,  denote an arbitrary permutation of 1, 2, ...,  n  and let  q(i),  for 0  i  n-1, denote an arbitrary permutation of 0, 1, ...,  n-1. We can define  p(i)  and  q(i)  for all integers  i  by assuming that p(i) = p(k)  and  q(i) = q(k)  whenever  is "k  (mod  n)       (4) Now consider the square arrays S and T defined by S(i,  j)  =  p(j +2mi)          (5a) T(i,  j)  =  q(i+2mj)...

  Narayan Pandit -General Algebraic framework - AI commented  We shall now see how the examples given by Narayana are particular instances of the above result. If we set  n =  4 and choose p(0) = 2, p(1) = 1, p(2) = 3, p(3) = 4, q(0) = 3, q(1) = 2, q(2) = 0 and q(3) = 1, we see that the conditions (6a) and (6b) are satisfied. The resultant 4x4 array S as defined by (5a) will be which is same as the  chadya  square considered by Narayana as shown in Fig. 1. The array 4x4 array 4T as defined by (5b) is given by which is nothing but the  chadaka  square considered by Narayana as shown in Fig. 1, except that the order of columns is reversed. Thus the composition S+4T is nothing but the folding process of Narayana as shown in Fig. 2 and leads to the pan-diagonal 4x4 magic square as shown in the Figure. In the same way, if we set  n =  8 and choose p(0) = 4, p(1) = 3, p(2) = 2, p(3) = 1, p(4) =  5, p(5) =  6, p(6) = '7,  p('7)=...

  Narayan Pandit - 7x7 Magic Square - AI Commented NARAYANA'S FOLDING METHOD FOR THE CONSTRUCTION OF  VISAMA  MAGIC SQUARES Narayana  has also outlined a method for constructing magic squares of odd orders by using the technique of folding two squares which are constructed suitably. His description of the procedure is as follows: Two sequences referred to as the  mulapankti  and the  gunapankti are  to be determined as earlier. The first number should be written in the middle cell of the top row and below this the numbers of the sequence in order. The rest of the numbers are to be entered in order from above. The first number of the second sequence is to be written in the same way [in the middle cell of the top row]; the second etc. numbers are also to be written in the same way. The rule of combining the covered and the coverer is the same as before. The details of this method are best illustrated by considering the following example discussed by...

  Narayan Pandit- Odd Order Magic Squares - AI Commented We shall now show that, Narayana's folding method is indeed a general procedure, which can be used to construct a large class of pan-diagonal  Visama  magic squares (magic squares of odd order). Fig. 7.  Folding Process for Construction of 7x7 Magic Square with Sum 238 Theorem 2: 3  Let  n  be an odd number and let  p(i)  for 1  i n  denote an arbitrary permutation of 1, 2, ...,  n  and let  q(i)  for 0  i  n-1 denote an arbitrary permutation of 0, 1, ..., n-1. We can define  p(i)  and  q(i)  for all integers  i  by assuming that p(i) = p (k)  and  q(i) = q(k)  whenever  is "k  (mod  n)      (12) Now consider the square arrays S and T defined by S(i,  j)  =  p(i+j)    (13a) T(i,  j)  =  q(i j)     (1...

  Narayan Pandit - 7x7 Magic Square by Folding Method - AI commented  We shall now see how the example given by Narayana is a particular instance of the above result. If we set  n =  7 and choose p(0) = 5, p(1) = 6, which is same as the  chadya  square considered by Narayana as shown in Fig. 6. The array 7x7 array 10T as defined by (13b) is given by which is nothing but the  chadaka  square considered by Narayana as shown in Figure 6, except that the order of columns is reversed. Thus the composition S+10T is nothing but the folding process of Narayana as shown in Fig.7 and leads to the 7x7 magic square as shown in the Figure. As we noted earlier, the 7x7 magic square in Fig. 7 obtained by Narayana's folding method is not pan-diagonal. We shall now show that, for the case of odd-numbers  n  which are not multiples of 3 (that is, numbers of the form 6m±1), a simple modification of Narayana's folding method can be used to construct pan-dia...