*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:

- Inder J. Taneja, Universal Magic Squares of Orders 128, 126 and 120 With Digits 1 and 8,
**Zenodo**, October 26, 2020,**http://doi.org/10.5281/zenodo.4130393**, pp. 1-194 - Inder J. Taneja, Universal Magic Squares of Orders 128, 126 and 120 With Digits 2 and 5,
**Zenodo**, October 31, 2020,**http://doi.org/10.5281/zenodo.4148929**, pp. 1-194 - Inder J. Taneja, Upside-Down Magic Squares of Orders 128, 126 and 120 With Digits 6 and 9,
**Zenodo**, October 31, 2020,**http://doi.org/10.5281/zenodo.4167058**, pp. 1-194.

**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

- Inder J. Taneja, Universal Magic Squares of Type 4k, 6k and 12k Using the Digits 1 and 8, Zenodo, June 28, 2020,
**http://doi.org/10.5281/zenodo.3911452**, pp. 1-134. - Inder J. Taneja, Universal Magic Squares of Type 4k, 6k and 12k Using the Digits 2 and 5, Zenodo, June 28, 2020,
**http://doi.org/10.5281/zenodo.3911457**, pp. 1-133. - Inder J. Taneja, Upside-Down Magic Squares of Type 4k, 6k and 12k Using the Digits 6 and 9, Zenodo, June 28, 2020,
**http://doi.org/10.5281/zenodo.3911461**, pp. 1-135

## Part III: Up to Orders 32

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

- 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:

- 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**. - Inder J. Taneja, Universal Magic and Bimagic Squares of Orders 17 to 32 With Digits 1 and 8, Zenodo, May 30, 2020, pp. 1-103,
**http://doi.org/10.5281/zenodo.3866366**

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:

- 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**. - Inder J. Taneja, Universal Magic and Bimagic Squares of Orders 17 to 32 With Digits 2 and 5,
**Zenodo**, May 30, 2020, pp. 1-113,**http://doi.org/10.5281/zenodo.3866386**

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:

- 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**. - 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**

## SUMMARY

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

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

In each group there are three papers:

- For digits 1 and 8. In this case the magic squares studied are
**upside-down**and**mirror looking**, i.e.,**universal**. - 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. - 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

**1024 Pandiagonal Magic Squares of Order 4 With Equal Magic Sums**

**441 Magic Squares of Order 6 With Equal Magic Sums**

**100 Pandiagonal Magic Squares of Order 12 With Equal Magic Sums**

Blocks of Order 3 are Semi-Magic Squares With Different Sums

Blocks 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:

For more and different procedures please download the papers.

The magic sums of each sub-magic squares multiples of order 4 is given by

## Slides for the Part II

**256 Pandiagonal Magic Squares of Order 4 With Equal Magic Sums**

**100 Magic Squares of Order 6 With Equal Magic Sums**

**25 Pandiagonal Magic Squares of Order 12 With Equal Magic Sums**

Blocks of Order 3 are Semi-Magic Squares With Different Sums

Blocks of Order 3 are Semi-Magic Squares With Different Sums

Categories: Magic Squares