New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108

Never seen before.

By boarded magic squares we understand that if we remove the external border still we are left with magic squares of lower orders with sequential entries. Single digits bordered magic squares are well-known. This work is for double digits not non in the literature. We are bringing this concept for the first time in the literature. During these days some studies in this directions are made. These are given below in seperate parts. One for even number orders and another for odd number orders. Most of the work is done manually. A special trick is applied to bring all these magic squares.

Lower order double digits magic squares can also be calculated using software by Harry While: Downloads (budshaw.ca).

Double Digits Bordered Magic Squares:
Even Number Orders

The work for order of type 4k, k=2, 3, … i.e., for orders 8, 12, 16, 20, 24, 28, 32, 36 and 40:
- Inder J. Taneja, Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24, Zenodo, April 26, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7866956.
  Site Link: Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24 .
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 28 and 32, Zenodo, April 26, 2023, pp. 1-36, https://doi.org/10.5281/zenodo.7866981.
  Site Link: Two Digits Bordered Magic Squares of Orders 28 and 32.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 36 and 40, Zenodo, May 04, 2023, pp. 1-41, https://doi.org/10.5281/zenodo.7896709.
  Site Links:
  1. Two digits Bordered Magic Squares of Order 36.
  2. Two digits Bordered Magic Squares of Order 40 .
The work is for the order of type 4k+2, k=2, 3, …, i.e., for the orders 10, 14, 18, 22, 26, 30, 34 and 38:
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 10, 14, 18 and 22, Zenodo, April 30, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7880931.
  Site Link: Two digits Bordered Magic Squares of Orders 10, 14, 18 and 22.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 26 and 30, Zenodo, April 30, 2023, pp. 1-45, https://doi.org/10.5281/zenodo.7880937.
  Site Link: Two digits Bordered Magic Squares of Orders 26 and 30.
- Inder J. Taneja, Two digits Bordered Magic Squares of Orders 34 and 38, Zenodo, May 10, 2023, pp. 1-45, https://doi.org/10.5281/zenodo.7922571.
  Site Links:
  1. Two digits Bordered Magic Squares of Order 34.
  2. Two digits Bordered Magic Squares of Order 38 .

Double Digits Bordered Magic Squares:
Odd Number Orders

The work for double digits odd order bordered magic squares:
For the previous work on odd orders from 3 to 31 see the following links:
- Inder J. Taneja, Odd Order Magic Squares: Orders 3 to 15, Zenodo, June 15, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8043030.
  Site Link: Odd Order Magic Squares: Orders 3 to 15
- Inder J. Taneja, Magic Squares of Orders 17 and 19, Zenodo, June 15, 2023, pp. 1-38, https://doi.org/10.5281/zenodo.8043105.
  Site Link: Magic Squares of Orders 17 and 19
- Inder J. Taneja, Magic Squares of Orders 21 and 23, Zenodo, June 15, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8043198.
  Site Links: Magic Squares of Order 21.
- Inder J. Taneja, Magic Squares of Orders 21 and 23, Zenodo, June 15, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8043198.
  Site Links: Magic Squares of Order 23.
- Inder J. Taneja, Magic Squares of Order 25, Zenodo, June 15, 2023, pp. 1-27, https://doi.org/10.5281/zenodo.8043228.
  Site Links: Magic Squares of Order 25.
- Inder J. Taneja, Magic Squares of Order 27, Zenodo, August 06, 2023, pp. 1-32, https://doi.org/10.5281/zenodo.8218291.
  Site Links: Magic Squares of Order 27.
- Inder J. Taneja, Magic Squares of Order 29, Zenodo, August 06, 2023, pp. 1-30, https://doi.org/10.5281/zenodo.8218771.
  Site Links: Magic Squares of Order 29.
- Inder J. Taneja, Magic Squares of Order 31, Zenodo, August 06, 2023, pp. 1-35, https://doi.org/10.5281/zenodo.8219053.
  Site Links: Magic Squares of Order 31.

The above work is done as extended study for each order. This work of double digits magic squares just given one example in each case for orders 7 to 108. See below few examples.

In the end excel files of completed work are also attached for download.