RECREATIONAL MATHEMATICS * MAGIC SQUARES

Water Reflection, Upside-Down & Mirror Looking Magic Squares Generator

Order 3–25. Choose the square order, the allowed digital digits, and the number of digits in every cell. Choose water reflection, upside-down, or all three: water reflection, upside-down and mirror looking. The results verify the selected row, column and diagonal conditions; Excel places the original and transformed squares side by side.

Created by Inder J. Taneja

Magic-square order

Digits in each cell

Maximum results

Magic condition

Transform

0, 1, 8 stay the same in a mirror; 2 and 5 swap with each other. (3, 4, 6, 7, 9 have no real mirror image.)

Digits to use

Zero normally appears only in an interior position. For 7×7 two-digit Upside-down squares, fixed-width entries such as 01 are preserved so all 49 combinations are available.

References — click to show or hide

Based on the author’s following research works:

Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 3 to 6, Zenodo, January 07, 2025, pp. 1–93, https://doi.org/10.5281/zenodo.14607070.
- Site link: Universal and Upside-Down Magic Squares of Order 3 to 6 (new site)
- Site link: Water Reflexive Magic Squares: Order 3 to 6 (new site)
- Site link: Universal and Upside-Down Magic Squares of Order 3 to 6 (old site)
- Site link: Water Reflexive Magic Squares: Order 3 to 6 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 7 to 10, Zenodo, January 07, 2025, pp. 1–171, https://doi.org/10.5281/zenodo.14607071.
- Site link: Universal and Upside-Down Magic Squares of Orders 7 to 10 (new site)
- Site link: Water Reflexive Magic Squares: Order 7 to 10 (new site)
- Site link: Universal and Upside-Down Magic Squares of Orders 7 to 10 (old site)
- Site link: Water Reflexive Magic Squares: Order 7 to 10 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 11 to 13, Zenodo, January 15, 2025, pp. 1–146, https://doi.org/10.5281/zenodo.14649320.
- Site link: Universal and Upside-Down Magic Squares of Orders 11 to 15 (new site)
- Site link: Water Reflection Magic Squares: Order 11 to 13 (new site)
- Site link: Universal and Upside-Down Magic Squares of Orders 11 to 15 (old site)
- Site link: Water Reflection Magic Squares: Order 11 to 13 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 14 to 16, Zenodo, January 15, 2024, pp. 1–140, https://doi.org/10.5281/zenodo.14649519.
- Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 16 (new site)
- Site link: Water Reflection Magic Squares: Order 14 to 16 (new site)
- Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 16 (old site)
- Site link: Water Reflection Magic Squares: Order 14 to 16 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 17 to 20, Zenodo, January 17, 2025, pp. 1–86, https://doi.org/10.5281/zenodo.14676293.
- Site link: Universal and Upside-Down Magic Squares of Order 20 (new site)
- Site link: Water Reflection Magic Squares: Orders 17 to 20 (new site)
- Site link: Universal and Upside-Down Magic Squares of Order 20 (old site)
- Site link: Water Reflection Magic Squares: Orders 17 to 20 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 21 to 23, Zenodo, January 20, 2025, pp. 1–83, https://doi.org/10.5281/zenodo.14688709.
- Site link: Universal and Upside-Down Magic Squares of Order 21 (new site)
- Site link: Water Reflection Magic Squares: Orders 21 to 23 (new site)
- Site link: Universal and Upside-Down Magic Squares of Order 21 (old site)
- Site link: Water Reflection Magic Squares: Orders 21 to 23 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 24, Zenodo, January 20, 2025, pp. 1–150, https://doi.org/10.5281/zenodo.14700325.
- Site link: Universal and Upside-Down Magic Squares of Order 24 (new site)
- Site link: Water Reflection Magic Squares: Order 24 (new site)
- Site link: Universal and Upside-Down Magic Squares of Order 24 (old site)
- Site link: Water Reflection Magic Squares: Order 24 (old site)
Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 25, Zenodo, January 21, 2025, pp. 1–79, https://doi.org/10.5281/zenodo.14715162.
- Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 25 (new site)
- Site link: Water Reflection Magic Squares: Order 25 (new site)
- Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 25 (old site)
- Site link: Water Reflection Magic Squares: Order 25 (old site)