Semi-Magic Knight's Tours


Links to related pages: Part 1: 4 by N Boards.Part 2: 6 by N Boards.Part 3: 8 by N and Larger Boards.


Introduction

This section of the notes has expanded rapidly and is now split into three parts with this page as an introduction. A semi-magic tour is a rectangular tour that adds to a constant sum in every line in one direction, but not in every line in the perpendicular direction. A quasi-magic tour is a semi-magic tour in which the non-magic lines add to only two different values. A near-magic tour is a semi-magic tour in which the perpendicular lines add to the magic constant and two other values. It is natural to study tours of these types in the case of boards that do not admit tours to be magic in both ranks and files, or on which no such magic tours have been constructed.

Part 1 details results on boards 4×6, 4×8 and 4×12, now including 48 quasimagic 4×12 tours found by Jean-Charles Meyrignac.

Part 2 by Awani Kumar, details results on the 6×6 board. To these results are now added five quasimagic 6×8 tours by Jean-Charles Meyrignac and two examples 6×12 by George Jelliss.

Part 3 covers examples on larger boards, including the 10×10 tour with quaternary symmetry found by Tom Marlow.

Oblong tours are oriented with the longer sides horizontal. Square tours have the magic lines horizontal. The tours are mostly oriented to have the smallest number in the top left corner, but those found by JCM have the number 1 in the first half of the top rank. The reverse numbering of a semimagic tour is also semimagic (and the same is true for quasimagic and near-magic tours).


The question remains; are quasi-magic knight tours on boards 4m by (4n + 2) such as the 8×10 and 12×6 the best that can be achieved or are magic knight tours possible on at least some boards of these proportions?

Magic rectangles can of course be constructed on boards 4m by 4n with m and n greater than 1, so we don't seek semi-magic examples on these boards unless they have other special properties.


Note added May 2004: It is also possible to form semi-magic tours on rectangles with one even side and one odd side. This seems not to have been studied in detail. Here is an example, from L'Echiquier, October 1929.
13 24  3 18  7 28  9
 2 21 14 25 10 17  6
23 12 19  4 15  8 27
20  1 22 11 26  5 16

The tour has rotational symmetry and the files all add to 58.


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