Proving the loneliness of the Khajuraho square

摘要

The two 4 × 4 matrices below have a lot in common. They both contain the numbers from 1 to 16 and, more remarkably, the numbers in each row, in each column and in the two main diagonals all add up to 34. The left matrix appeared around 1100 AD in an inscription in the Parshvanath Jain temple in Khajuraho. Therefore we call it the Khajuraho square. The right matrix came to fame due to its occurrence in Albrecht Dürer's engraving Melancholia I. In the Khajuraho square not only the sum of the numbers on the two main diagonals, but also the sum of the numbers on each of the six broken diagonals, is equal to 34. For instance we have: 2 + 12 + 15 + 5 = 34, 16 + 13 + 1 + 4 = 34 or 12 + 8 + 5 + 9 = 34. Dürer's square on the other hand lacks this property. Are the conditions stated above, including the one about the broken diagonals, possibly sufficient to characterise the Khajuraho square?