# Upside-Down Magic and Bimagic Squares of Orders 3 to 32 Using 6 and 9

This work brings, magic squares of order 3 to 32 just with two digits in such a way that the magic squares are upside-down independent of magic sums. This is done only with two digits 6 and 9. The works for the digits 1 and 8, and 2 and 5 are given separately. In case of 2 and 5, the numbers are written in digital form. In mirror looking case, 2 becomes 5 and 5 as 2. For the case of 6 and 9, we have only upside-down magic squares. In this case, when making 180 degrees rotation 6 becomes 9 and 9 as 6. In case of order 3 it is semi-magic square. The work is divided in two papers. One for the orders 3 to 16 and another for orders 17 to 32. Below are links for download:

1. Inder J. Taneja, 2-Digits Universal and Upside-Down Palindromic Magic and Bimagic Squares: Orders 3 to 16, Zenodo, April 07, 2020, pp. 1-103, http://doi.org/10.5281/zenodo.3743362.
2. Inder J. Taneja, Upside-Down Magic and Bimagic Squares of Orders 17 to 32 With Digits 6 and 9, Zenodo, May 30, 2020, pp.1-98,  http://doi.org/10.5281/zenodo.3866396

The first link is for all the three cases, while the second link in only for the digits 6 and 9. The magic squares of orders 8, 9, 16, 25 are also written in bimagic squares. While for the case of order 24, the result is semi-bimagic. The block-wise constructions of magic squares of orders 8, 9, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30 and 32 are also given. The cases of orders 15, 21 and 27 are semi-magic squares. The whole the work is without use of any programming language. The work is done just with number’s combinations. For the orders 3 to 16 the palindromic magic squares are also given.

There are 4-digits in each cell for the magic squares of orders 3 and 4. 6-digits in each cell for the magic squares of orders 5 to 8. 8-digits in each cell for the magic squares of order 9 to 16. 10-digits in each cell for the magic squares of order 17 to 32. For the semi-bimagic and bimagic squares of orders 24, 25 and 32, there are 12-digits in each cell.

Similar kind of work for the order 33 to 64 (12-digits in each cell) , and for the orders 65 to 128 (14-digits in each cell) is under preparation.

Examples: