Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 12
Inder Jeet Taneja · Zenodo (CERN European Organization for Nuclear Research) · 2021
During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 108. Based on these two big magic squares, the inner order magic squares multiples of 12 are studied. By inner orders we understand as the magic squares of orders 96, 84, 72, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 12, 24, 36, etc. The construction of the block-wise bordered magic squares multiples of 12 is based on equal sum blocks of magic squares of order 12. It is done in six different ways. First three ways are such that each magic square of order 12 is composed by blocks of orders 3, 4 and 6. The forth way is bordered magic squares. Two blocks of order 12 composed with small blocks of order 3 and 4 are pandiagonal. This lead us to write all orders multiples of 12 as pandiagonal magic squares. The only difference is that the pandiagonal magic squares multiples of 12 are no more block-wise bordered magic squares. Moreover, the magic squares from orders 36 onwards are block-wise bordered magic squares. The advantage in studying block-wise bordered magic squares is that when we remove external borders, still we left with magic squares with sequential entries. The bordered magic squares also have the same property. The difference is that instead of numbers here we have blocks of equal sum magic squares multiples of 12. For multiples of orders 4, 6, 8 and 10, see author's recent works (multiples-4, multiples-6, multiples-8 and multiples-10). The further multiples, such as multiples, for order 14, shall be done in another works. This work brings examples only up to order 48. Higher order examples are given in Excel file attached with the work.