ABSTRACT
We now consider the set of all 4 × 4 pan-diagonal magic squares with entries given by the first sixteen natural numbers 1, 2, . . . , 16. We note first that in this case, the sum of the entries in any row, column, diagonal or broken diagonal is 34. In his Gan. itakaumudi¯ (c.1356 CE), Na¯ra¯yan. a Pan.d. ita displayed 24 pan-diagonal 4×4 magic squares, with different cells being filled by different numbers from the arithmetic sequence 1, 2, . . . , 16, the top left entry being 1. Na¯ra¯yan. a also remarked that (by permuting the rows and columns cyclically) we can construct 384 pan-diagonal 4 × 4 magic squares with entries 1, 2, . . . , 16. In 1938, Rosser and Walker (1938), proved that this is in fact the exact number of 4 × 4 pan-diagonal magic squares with entries 1, 2, . . . , 16. Vijayaraghavan (1941), gave a much simpler proof of this result, which we shall briefly outline here. We begin with the following Lemmas due to Rosser and Walker:
Lemma 1. Let M be a pan-diagonal 4 × 4 magic square with entries 1, 2, . . . , 16, which is mapped on to the torus by identifying opposite edges of the square. Then the entries of any 2 × 2 sub-square formed by consecutive rows and columns on the torus add up to 34.
