Bimagic Queen's Tours and a Semi-Trimagic Queen's Tour on a 16x16 Board

Awani Kumar, a specialist of magic chess tours, and in particular the knight's tour, recently wrote an email to a magic square group that I belong to, with the following message:

"I have a humble request/suggestion. We have bimagic, trimagic and higher order magic squares but the bimagic knight tour has remained elusive.

Please look for bimagic knight tour on say, 32x32 or 64x64 (or larger boards). It need not be diagonally magic. Even a semi-bimagic tour -- only rows or columns but not both -- will be a milestone."

Bimagic Queen's Tours Research and Findings on a 16x16 Board

I had no bimagic knight's tours to propose, but Awani Kumar's request reminded me of research I once did on bimagic queen's tours, back in 2021 and 2022. I had then been hoping to find a diagonally bimagic queen's tour on the 16x16 board, but did not succeed. Dissatisfied, I never published any of my findings, but after reading Awani's message I have since decided to compile these in the following PDF document entitled "Bimagic Queen's Tours and a Semi-Trimagic Queen's Tour on a 16 x 16 board." The document can also be found at https://drive.google.com/file/d/1SobJ0HOpMgTbe6eiA_k2-Cx8pBU2o6LI/view?usp=sharing.

A Single-Break Diagonally Bimagic Queen's Tour 

This single-break diagonally magic queen's tour was found on the 28th August, 2021, but remained unpublished until now. The single break occurs between the numbers 128 and 129, at exactly the half-way point between 1 and n² (where n = 16, and n² = 256).

Observations

I'm sure that break-free diagonally magic queen's tours exist on the 16 x 16 board. Should the subject be of interest to you, I look forward to reading about your findings!

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Multiplicative Magic Tori

Because the present-day definition of complementary and self-complementary magic squares can be over-restrictive when studying magic tori, an alternative modular arithmetic interpretation is therefore proposed.

This approach suggests a new direction for research on magic squares (or magic tori), using modular multiplication, modular addition, and modular exponentiation.

Multiplicative Magic Tori (MMT) are examined throughout the orders 1 to 4. (Although there is no magic torus of order-2, there is a Multiplicative Diagonally Semi-Magic Torus (MDSMT)). The study then continues with a partial inspection of various examples from higher-orders, including some bimagic MMT of order-8.

A typical Multiplicative Magic Torus (MMT), or Multiplicative Magic Square (MMS) of order-4

The results include a new census of the Multiplicative Magic Tori (MMT) and Multiplicative Magic Squares (MMS) of orders 1 to 4. A detailed classification of the 82 Multiplicative Magic Tori (MMT) and 220 Multiplicative Magic Squares (MMS) of order-4 is given, together with explanatory graphics that highlight the main relationships and links.

In addition, it is shown that the diagonally magic, diagonally semi-magic, and lesser-magic tori, which are also present on the MMT, have special orthogonal sums. Some other interesting characteristics of modular exponentiations are also examined and commented.

The conclusions are presented in the form of integer sequences. Any input that might confirm, correct, or usefully append these findings, would be much appreciated

New Developments! 

Posted on the 24th April 2024, a new article entitled "Plus or Minus Groups of Magic Tori of Order 4" now demonstrates that the 255 magic tori of order 4 come from 137 ± Groups!

Although "Plus or Minus Groups of Magic Tori of Order 4" proposes an alternative way of looking at things, the "Multiplicative Magic Tori" represents valid intermediary research, and therefore remains available below.

Multiplicative Magic Tori (outdated by findings in the 137 ± Groups)

Please note that if you click on the button that appears at the top right hand side of the pdf viewer below, a new window will open and full size pages of "Multiplicative Magic Tori" will then be displayed, with options for zooming.

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