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The 3 by 3 diagonal magic square was known in China about 2200BC. All eight sets of three numbers in line add to the same magic constant, 15.
If the successive numbers are joined by straight lines the result is a tour of the 3 by 3 board using moves with coordinates {0,1), {1,1}, {1,2} which define the chessmen wazir, fers and knight. It can be regarded as a tour by a centaur which moves in all three ways. |
All diagonal magic squares 4 by 4 were listed by B. Frénicle de Bessy. (He died in 1675, and his results were published in 1693.)
Only one uses two types of move, namely this knight + wazir (= emperor) tour:
This has axial symmetry, whereas the 3 by 3 tour has central symmetry. These geometric properties result in the arithmetic property that numbers equidistant on opposite sides of the axis or centre add to the same number (n^2 + 1 on n by n boards). |
This knight's tour of the 5 by 5 board by Leonhard Euler (1759) is a figured tour since it has various arithmetical properties: in particular the numbers 1, 7, 13, 19, 25 on the diagonal form an arithmetic progression with common difference 6.
However it is not a magic tour since only the four sets of five numbers on the lines through the centre cell add to 65: other ranks and files do not. This partial magic property is due to the tour's centro-symmetry: numbers in opposite cells adding to 26. |
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There are just 5 distinct closed knight tours with quaternary symmetry on the 6×6 board.
Found by Paul de Hijo (Phillipe Jolivald) 1882 and independently rediscovered many times since.
There are 17 tours 6 by 6 with binary symmetry (shown by M. Kraitchik 1927) and 1223 asymmetric, making
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The first magic knight's tour, found by William Beverley, was published in 1848. It is shown here in both arithmetical and geometric forms.
Work by G. Stertenbrink, J-C. Meyrignac and H. Mackay in 2003 has shown that there are 140 distinct arithmetical magic knight tours on the 8 by 8 board (not counting reversals, rotations and reflections). None of these tours has a magic diagonal. The work is reported at: Computing Magic Knight Tours. |
The simplest of the four diagonal magic knight tours on the 12 by 12 board found by Awani Kumar in 2003.
When the cells are numbered, all ranks and files and both diagonals add to 870. |
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The smallest rectangle that admits a knight's tour is the 3 by 4. There are three geometrically distinct open tours (four arithmetically, since the asymmetric tour, numbered from each end, gives two arithmetical forms). These were found by Euler (1759).
The smallest rectangular boards allowing closed tours by the knight are the 30-cell boards 3 by 10 and 5 by 6. There are six on the 3 by 10 (first found by E. Bergholt 1917 and G. L. Moore 1920): two are centrosymmetric tours (of bergholtian type in which two moves cross at the centre), two axisymmetric (of sulian type in which the axis does not pass through any cell-centre) and two asymmetric.
There are only three on the 5 by 6: One asymmetric and two axi-symmetric (sulian type).
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Mediaeval manuscripts contain knight's tours on the 4 by 8 half-chessboard, the earliest being that by the Kashmiri poet Rudrata circa 900AD.
When numbered, the example by H. J. R. Murray is magic in the 4 ranks (sum 132) and in 6 files (sum 66), while mine is magic in the 8 files and in 2 ranks.
Tours on 4-rank boards were analysed by C. Flye Sainte-Marie 1877, showing that they must be open, start and finish in the outer ranks, and consist of two half-tours connected by a single move between the two inner ranks. He made a complete enumeration of the tours on the 4 by 8 board, finding 7772 distinct tours. |
