Puzzle 1141 Closed Magic Knight tours 12x12

Puzzle 1141 Closed Magic Knight tours 12x12
Carlos Rivera poses the following puzzle:

We have already dealt with the Knight tour problem at least two times: in our Puzzles 30 & 310.

Let's summarize the basic definitions (4) and know/published results (8):

Definition 1: The ‘knight's tour’ is a numbered tour of a knight over a otherwise empty nxn board, visiting each square only once. The numbers allocated in each square visited by the knigt must be numbered orderly from 1 to nxn. You are free to select the starting cell and the whole route.

Definition 2: If the first cell and the last cell of the knight tour is separated by a simple knight movement the tour is called "closed or re-entrant", otherwise the tour is called "open or not-closed or non-re-entrant".

Definition 3: The knight tour is called "Magic Knight Tour" if the numbers allocated in the matrix nxn produces a Magic square; otherwise it is called simply "Knight tour"

Definition 4: A "Magic square" is a numbered matrix in which the sum of each one of the the rows , each one of all the the columns and the two main diagonals is the same quantity. If this condition fails for one or the two main diagonals the matrix is called "semi magic".

Main results, for our purposes:

1) We know closed Knight tours 8x8 since year 840 (Al Adli) and year 910 (Al-Suli).
See the following Fig 1 and Fig 2

2) In 1823 H. C. von Warnsdorff provided the first heuristic rule to get a Knight Route. "The knight is moved so that it always proceeds to the square from which the knight will have the fewest onward moves. When calculating the number of onward moves for each candidate square, we do not count moves that revisit any square already visited. It is possible to have two or more choices for which the number of onward moves is equal; there are various methods for breaking such ties, including one devised by Pohl[21] and another by Squirrel and Cull." Cfr. https://en.wikipedia.org/wiki/Knight%27s_tour

Other methods dedicated exclusively to the 8x8 knight tours are:

a) De Moiver & MontMort (before 1719, but published by W.W. Rouse Ball until 1892)
b) L. Euler (1766)
c) Peter Margk Roget

Cfr. https://culturacientifica.com/2022/10/19/el-problema-del-recorrido-del-caballo-en-el-tablero-de-ajedrez-ii/

3) As far as is known, William Beverly in 1848 tried without success to produce a magic knight tour, 8x8. His best was an open semi-magic knight tour, 8x8. Later, in 1849 Carl Wenzelides got a closed semi-magic knight tour, 8x8.

See the following Fig 3 and Fig 4.

4) Some others as Harold J. R. Murray (1868 – 1955), Thomas W. Marlow (1927-2011) y Timothy S. Roberts (2003) produced a lot more of 8x8 semimagic tours open or closed.

5) In 2003, Hugues Mackay (Canada), Jean-Charles Meyrignac (Francia) and Günther Stertenbrink (Alemania) produced ALL the possible Semi magic Knight 8x8 tours (140, 63 closed and 77 open) with the surprising result that NONE of them resulted to be "magic". All of these were semi magic.

6) For squares nxn other than 8x8, it was demonstrated that mo magic or semi-magic tours are possible for n odd. Later (2003) G. P. Jelliss demonstrated that the same occur for n even of the type 4k+2. So the only possibility of obtaining magic or semi magic knight tours are for these squares n=4k. But for magic tours it has been shown that are impossible for squares 4x4 and 8x8. So the first possibility open is 12x12 or greater.

For notes 1, 3, 4, 5 and 6 Cfr. https://culturacientifica.com/2022/11/02/existen-recorridos-magicos-del-caballo-en-el-tablero-de-ajedrez/#:~:text=El%20resultado%2C%20obtenido%20en%202003,caballo%20que%20generen%20cuadrados%20m%C3%A1gicos!

7) George Jelliss in his page (https://www.mayhematics.com/t/md.htm) about 12x12 Magic Knight Tours writes that:
"At this very moment (August 2023) there are known only four (4) Magic Knigt Tours in a 12x12 matrix, by Awani Kumar, 2003. But all of them are "not closed tours".

See one of these four in the following Fig 5

8) Open Problem: Are there Magic Knight 12x12 Tours of the closed form?

Q. Can you produce one Magic Knight 12x12 Tours of the closed form, If so, show one of them? Or show that there are none of them.