# 2-Digits Universal Magic Squares Up To Orders 128

The whole work is without use of any programming language.
It is just based on the number’s combinations
.

## Part I: Orders 128, 126 and 120

This work brings magic squares multiples of 4, 6 and 12 using only two digits: (1,8), (2,5) and (6,9). Each magic square contains 14-digits in each cell. Just with two digits, one can have exactly 16384 different possibilities of 14-digits combinations. In case of multiple of 4, we have written magic squares of orders, 4, 8,…., 60, 64,…, 124, 128. In case multiple of 6, we have written magic squares of orders 6, 12, …, 54, 60, 120, 126. In case multiple of 12, we have written magic squares of orders 12, 24, 36, 48 , 60, 72, 84, 96, 108 and 120. In each case, all the blocks of magic squares of orders 4, 6 and 12 are with equal magic sums. In case of digits (1,8) and (2,5), the magic squares are upside-down and mirror looking. The digits 2 and 5 are used in digital forms. In case of digits 6 and 9, the magic squares are only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. Below is a link of these three works for download:

In the end there ares slides of orders 4, 6 and 12. Also there are procedures to construct sub-orders magic squares.

## Part II: Orders 64 and 60

This work brings magic squares multiples of 4, 6 and 12 using only two digits: (1,8), (2,5) and (6,9). Each magic square contains 12-digits in each cell. In case of multiple of 4, we have written magic squares of orders, 4, 8,…., 60, 64. In case multiple of 6, we have written magic squares of orders 6, 12, …, 54, 60. In case multiple of 12, we have written magic squares of orders 12, 24, 36, 48 , 60. In each case, all the blocks of magic squares of orders 4, 6 and 12 are with equal magic sums. In case of digits (1,8) and (2,5), the magic squares are upside-down and mirror looking. The digits 2 and 5 are used in digital forms. In case of digits 6 and 9, the magic squares are only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. Below are links of these three works for download

## Part III: Up to Orders 32

For the work up to order 32 in all the three possibilites see the link below:

1. Inder J. Taneja, Universal Magic and Bimagic Squares of Orders 3 to 32 Using 1 and 8, https://inderjtaneja.com/2020/06/09/universal-magic-and-bimagic-squares-of-orders-3-to-32-using-1-and-8/

It contains following two works:

2. Inder J. Taneja, Universal Magic and Bimagic Squares of Orders 3 to 32 Using 2 and 5, https://inderjtaneja.com/2020/06/09/universal-magic-and-bimagic-squares-of-orders-3-to-32-using-2-and-5/

It contains following two works:

3. Inder J. Taneja, Upside-Down Magic and Bimagic Squares of Orders 3 to 32 Using 6 and 9, https://inderjtaneja.com/2020/06/09/upside-down-magic-and-bimagic-squares-of-orders-3-to-32-using-6-and-9/

It contains following two works:

## SUMMARY

Summarizing all the three Parts, the work contains total 12 papers divided in following groups:

1. Magic square up to Orders 16. It has total 8-digits in each cell.
2. Magic square up to Orders 32. It has total 10-digits in each cell.
3. Magic square up to Orders 64. It has total 12-digits in each cell.
4. Magic square up to Orders 128. It has total 14-digits in each cell.

In each group there are three papers:

1. For digits 1 and 8. In this case the magic squares studied are upside-down and mirror looking, i.e., universal.
2. For digits 2 and 5. In this case the magic squares studied are upside-down and mirror looking, i.e., universal. For this we need to write digits written in digital form.
3. For digits 6 and 9. In this case, the magic squares are only upside-down.

Combining the above four groups with three papers each, lead us to 12 paper given above for download.

## Slides for the Part I

#### 100 Pandiagonal Magic Squares of Order 12 With Equal Magic SumsBlocks of Order 3 are Semi-Magic Squares With Different Sums

One of the procedure used for sub-magic squares of order 4 is based on the following figure: