Here we survey results on 20×20 and larger squares.
As for the 12 × 12 and 16 × 16 cases we show the tours graphically, leaving the reader to put in the numbers and check the magic totals if desired.
The Fairy Chess Review vol.9 no.5 August 1955 p.44 gave a brief obituary of H. J. R. Murray who died on 16th May that year aged 86.
Some results by Murray on larger boards using his Method of Extended Quartes are shown below from his unpublished manuscripts (1942 and 1951).
The next issue of FCR vol.9 no.6 October 1955 p.46-47 reported work by H. E. de Vasa and T. H. Willcocks based on Murray's method
to construct magic tours on boards of side 24, 32, 48 and 64, though regrettably space prevented publication of diagrams. Some results by de Vasa and Willcocks are included
below, from their correspondence with the editors of FCR (Dennison Nixon and C. E. Kemp) that were deposited in the Bodleian Library Oxford in 1991,
and some from my own correspondence with T. H. Willcocks around 1986, at the time of the special Issue #26 of Chessics on Magic Tours.
Magic tour, non-diagonal (Jelliss undated) by the split and fix method. The magic constant is 4010.
Here is a diagonally magic open tour 20×20 by T.H.Willcocks from Journal of Recreational Mathematics (1968). Both diagonals add to 4010. The central 4×4 is a double Beverley quad. The heavier lines mark the links between the quarters (100-101, 200-201, 300-301).
Awani Kumar published a number of 20×20 magic tours in various magazines during 2020, to celebrate the year. However he has now constructed the two diagonally magic examples shown in digrammatic form here. These were published in Best Problems (Rassegna dei miglioni problemi) produced by Antonio Garofalo for the API (Associazione Problemistica Italiana) Anno XXVI, No.104, October 2022, pages 559-560, where they are shown in numerical form.
The dots mark the cells numbered 1 (q2), 100 (c5), 101 (d3), 200 (r2), 201 (s4), 300 (a3), 301 (b5), 400 (t4) which are the ends of the quarters (the 20 files being lettered a - t). The ranks, files and diagonals all add to the magic constant 4010. One of the broken diagonals of the above tour, running from a19 - s1 and including the top right corner cell t20, also adds to 4010.
The dots again mark the cells numbered 1 (t15), 100 (c5), 101 (d3), 200 (s15), 201 (r13), 300 (a3), 301 (b5), 400 (q13) which are the ends of the quarters (the 20 files being lettered a - t). The ranks, files and diagonals all add to the magic constant 4010. This tour has the extra feature that the squares of the numbers in the top rank add to 1070670, which is the magic sum required for a 'bimagic square'. However this is the only bimagic row in this tour.
E. Lange 24×24 magic tour formed by placing a border of 4×8 components round a 16×16 tour, which is a bordered form of Beverley's tour. Diagonals add to 6924 – 222 and + 38.
T. H. Willcocks (1956) diagonally magic tour composed 16 Nov 1956.
This was a product of his work with H. E. de Vasa mentioned in Fairy Chess Review 1955.
It is based on the (00a) tour by Jaenisch (1862). It was sent to me in a handwritten letter around 1986.
H. J. R. Murray 32×32 open nondiagonal magic tour by the method of extended quartes. Every quarterboard and 4×4 quad is also magic. Any square combination of quads is magic. The parent 8×8 magic square is Beverley's (27a). The magic constant is 16400.
H. J. R. Murray 32×32 closed magic tour. Diagonals add to 8368 and 23920 (reverse 8880 and 24432). This is the last tour in the Murray 1951 ms (Fig.12.22). It is based on his own magic tour (12p). I also had a note of the same tour but in a different orientation (reflected in diagonal). The quarters are not identical, they reflect left to right, but not upper to lower.
H. E. de Vasa. First fully magic knight tour on the board of 32×32 squares. Sent to Dennison Nixon and C. E. Kemp in 1956 and mentioned in FCR. The parent tour is (00a) by Jaenisch 1862.
Here is a diagonally magic 32×32 open tour composed by T. H. Willcocks at the same time as the de Vasa tour (November 1956) as reported in FCR. The details of this were sent to me by Mr Willcocks in a handwritten letter. The the 8×8 parent tour is (34d) by Ligondes 1883, the same as in the 16×16 tour composed at the same time. The four quarters, apart from the linking moves shown by bolder lines, are all identical apart from orientation, being arranged in direct quaternary symmetry.
Helge Emanuel de Vasa diagonally magic tour 48×48, composed Feb 1957. This diagram shows one quarter of the tour, the other three quarters reflecting this one through the medians and centre. Numbering, 1 to 48² = 2304, starts at one circled dot and ends at the other. Magic constant 55320. The connections in the central 8×8 are the same as for tour (00a). Diagram used as cover illustration on Chessics #26 (1986) special issue on Magic Tours.
