I have decided [August 2022] that these notes, which look at a traditional subject from new directions, and date from May 2011, deserve a page of their own, which they did have at one time. However [December 2022] I have now made this page part of a 'Multimover' section. Only one of the 4×4 diagonal magic squares uses two move types.
Sections on this page:
— 4×4 Non-diagonal Magic Tours Biaxial Tours with Two Types of Move
— [4×4 Diagonally Magic Tours] Historical Notes
— The Magic Sets
— Types of Move Used
— Diagonal Magic Tours with Three Types of Move
— The Eightfold Magic Tours
— The Fourfold Magic Tours
— The Threefold Magic Tours
— The Twofold Magic Tours
— The Onefold Magic Tours
In Chessics (#26, 1986, p.119) I studied magic squares by double-pattern leapers that show biaxial symmetry when the two endpoints of the tour are joined up to give a closed path, finding 10. These are not diagonally magic. The last two diagrams show the same closed tour numbered from different points.
These tours will also be found diagrammed individually in the Emperor, Empress, Queen, and Double-Pattern Leaper pages.
Various manuscripts, by al-Beruni c.1200, include many 4×4 diagonally magic squares, using letters as numbers, with the top row spelling a word. [Singmaster 1991, pp.218–9].
Bernard Frénicle de Bessy, who died in 1675, was the first to list all 880 arithmetically distinct 4×4 diagonal magic tours. The work was not published until eighteen years after his death in 1693 [see the Biobibliography for fuller details]. A computer print-out of the list is given in New Recreations with Magic Squares by W. H. Benson and O. Jacoby (Dover Publications 1976). The numbers on the tour diagrams below refer to this listing.
When the diagonal magic tours are counted geometrically the number 880 reduces to 616 since 264 are reverse-numberings of others, the other 352 symmetric tours being their own reversals. If we put in the closure move, joining 16 to 1, the number of geometrically distinct tours reduces further to 382, because some of the closed tours can be numbered magically from different origins.
There are 86 magic sets of 4 of the numbers 1, 2, ..., 16 adding to the magic constant 34. Of these, 28 are combinations of pairs of complements: (1,16) (2,15) (3,14) (4,13) (5,12) (6,11) (7,10) (8,9). They are represented graphically by the central symmetrical diagram. Only 8 of these occur in satins from the natural numbering of the array: (1,6,11,16), (1,7,10,16), (2,5,12,15), (2,8,9,15), (3,5,12,14), (3,8,9,14), (4,6,11,13), (4,7,10,13), these are shown in red. Each of these self-complementary sets contains two small numbers and two large numbers, also two even and two odd numbers. Any set with these properties may be called balanced, high-low or odd-even.
The other 58 cases occur in 29 complementary pairs (shown in the left and right parts of the diagram) of which 16 pairs are balanced both ways. Half of these are the ones that occur in satins, listed here in pairs: (1,6,12,15) : (2,5,11,16), (1,7,12,14) : (3,5,10,16), (1,8,10,15) : (2,7,9,16), (1,8,11,14) : (3,6,9,16), (2,7,12,13) : (4,5,10,15), (2,8,11,13) : (4,6,9,15), (3,6,12,13) : (4,5,11,14), (3,8,10,13) : (4,7,9,14).
Of the 13 pairs that remain, 6 (shown green) are balanced high-low but not even-odd (1,5,13,15) : (2,4,12,16), (1,7,11,15) : (2,6,10,16), (3,5,11,15) : (2,6,12,14), (3,7,9,15) : (2,8,10,14), (3,7,11,13) : (4,6,10,14), (5,7,9,13) : (4,6,10,12), and 6 (shown yellow) are balanced even-odd but not high-low (1,9,10,14) : (3,7,8,16), (1,10,11,12) : (5,6,7,16), (2,9,10,13) : (4,7,8,15), (2,9,11,12) : (5,6,8,15), (3,9,10,12) : (5,7,8,14), (4,9,10,11) : (6,7,8,13). This leaves one pair (1,9,11,13) : (4,6,8,16), that is unbalanced (shown in blue at the bottom of the diagrams). It is possible that a different scheme of ordering the complementary pairs may produce a more regular pattern.
I set as a puzzle in The Games and Puzzles Journal in 1988 (problem in #5+6, p88; solution in #7, p.112) to identify the two unbalanced sets and to use them in numerical order in the diagonals of a magic square. Solutions are provided by tours 75, 79, 166, 167 in Frénicle's list, and many others use the same numbers but not in sequence. The 166 example is among the tours with three types of move illustrated below.
There are 9 types of move possible on the 4×4 board. I letter them WFDNATCZX corresponding to the variant chess pieces Wazir {0,1}, Fers {1,1}, Dababa {0,2}, Knight {1,2}, Alfil {2,2}, Threeleaper {0,3}, Camel {1,3}, Zebra {2,3} and Tripper {3,3}.
Very surprisingly all but one of the diagonal magic squares require three or more types of move; the single exception is number 100 in the list. This uses only wazir and knight moves (WN), and in variant chess terms is thus an emperor tour. This special case does not seem to have been noticed until I reported it in Chessics (#26, 1986, p.119).
The 36 magic tours that use three types of move are illustrated below. Where the ends of the tour can be joined by a move used elsewhere in the tour (i.e. the tour is reentrant) this closure move is put in. Tour 617 is related to the emperor tour 100 in this way, the closure move being a new type of move, a three-leap. Some closed tours give a second solution, or pair of solutions, when numbered from a different point in the circuit. These cases are shown by linked diagrams.
The WFT example can be regarded as a Queen tour. The 6 WDN examples can be regarded as Knighted Rook tours. The WNA and FDN examples can be regarded as Knighted Queen tours. The WNZ combination gives the most solutions and can be regarded as a {n, n−1} leaper for any n, or a rider that makes a wazir move followed by a bishop move (at an obtuse angle).
The most elaborate examples of closed tours that can be magically numbered from different origins are the 6 eightfold magic tours.
These are equivalent to 48 tours in Frénicle's list. Two of these six use four types of move, 89 FDNZ, 101 WDNZ, the others use five or six:
In these the cells occupied by the ranges 1-2-3-4, 5-6-7-8, 9-10-11-12, 13-14-15-16 are all diagonal satins, forming a latin square,
as are the progressions 1-5-9-13, 2-6-10-14, 3-7-11-15, 4-8-12-16.
The first square for example can be analysed as a combination of latin squares as follows:
1 6 15 12 0 1 3 2 1 2 3 4
11 16 5 2 = 4 x 2 3 1 0 + 3 4 1 2
8 3 10 13 1 0 2 3 4 3 2 1
14 9 4 7 3 2 0 1 2 1 4 3
In the diagrams above I show the ranges by continuous lines and the links between them as dashed lines. The dots mark the points where the numbers 1, 4, 5, 8, 9, 12, 13, 16 occur. The ranges form a symmetric pattern but the links break this symmetry. I give the number of the first example in the Frénicle list (as reproduced by Oswald and Jacoby).
For the record, I list here the Frénicle numbers of the octets. From each given tour three others are derived
by renumbering the first tour from 5, 9 and 13, and four others are the reversals of these:
(62,524,678,389;505,710,269,165), (65,473,682,281;508,623,274,104), (89,345,661,554;323,721,464,149),
(93,294,664,469;326,621,467,109), (101,384,622.515;280,705,471,159), (106,548,620,333;468,715,291,140).
There are 64 fourfold magic tours. These include 24 symmetric cases: 8 rotary (60,250,478,632), (66,377,511,695), (84,299,451,637), (95,328,539,698), (122,321,489,649), (126,308,504,669), (183,368,460,657), (185,264,535,674), and 16 axial: (49,256,532,691), (54,261,463,660), (57,247,521,708), (69,312,514,654), (74,365,456,690), (76,267,459,677), (83,341,450,716), (98,331,499,671), (119,351,488,723), (123,307,528,713), (136,317,543,648), (147,296,550,636), (154,379,503,668), (163,387,475,630), (171,355,498,670), (174,311,530,651). These occur in sets of four having the same quartes but differently linked. Because of the symmetry, numbering from the four dots on the right give the same tours as numbering from the four on the left. Those with four types of move are 60, 147 WFNC, 66 FDNC, 69 WDNZ, 76 DNTZ, as also are derived tours that delete a link that is the only move of its kind. The others all use five or six types of move.
There remain 40 asymmetric that are fourfold magic: (8,409;605,242), (10,411;607,244), (21,213;591,445), (25,217;590,444), (27,421;583,233), (31,420,582,237), (40,275;585,437), (41,276;687,496), (56,246;818,768), (61,251;817,766), (82,316;844,779), (85,322;843,778), (102,279;828,785), (103,282;827,784), (105,283;588,442), (107,292;839,788), (108,293;838,787), (114,481;642,301), (115,302;640,479), (116,304;647,485), (117,483;646,305), (118,509;644,272), (141,334;854,809), (142,335;733,568), (148,553;720,344), (160,385;855,810), (164,388;709,523), (169,356;852,797), (173,362;853,798), (191,572;735,399), (192,400;734,570), (193,559;729,406), (238,240;415,413), (268,300;455,453), (271,303;510,486), (340,343;546,544), (438,470;758,757), (439,472;795,786), (491,493;754,753), (519,522;814,800).
There are 16 threefold magic tours. These are axially symmetric tours which have one symmetric numbering which is its own reversal, and another pair of asymmetric numberings that are reversals of each other — the second symmetric numbering (using the link without dotted ends) still gives a magic square but it is non-diagonal; (38;652,655), (50;684,685), (78;319,650), (131;364,689), (143;717,700), (152;693,692), (259;492,494), (370;531,525), (454;526,407), (545;497,277), (624;724,202), (627;373,205), (639;711,208), (706;332,94), (719;688,42), (725;666,88). These occur in pairs. 152 may have fewest intersections; 205 uses only 3 types of move (WNZ), as does 131 in the symmetric open version.
There are 232 twofold magic tours. These consist of 96 axially symmetric when open: (12,218), (13,427), (14,430), (15,223), (16,435), (17,225), (18,226), (19,433), (44,595), (45,252), (46,756), (48,763), (51,760), (52,602), (53,258), (55,761), (70,452), (71,769), (73,774), (77,771), (80,772), (81,457), (86,615), (87,665), (91,667), (100,617), (110,791), (111,792), (121,490), (125,309), (127,739), (130,533), (132, 813), (133,801), (135,547), (137,806), (150,366), (151,816), (155,802), (156,751), (157,807), (158,383), (182,367), (184,534), (187,394), (188,574), (189,561), (190,398), (219,593), (220,755), (221,597), (222,764), (224,603), (227,600), (253,819), (255,824), (257,821), (262,822), (284,776), (285,793), (286,833), (287,836), (288,837), (295,634), (298,638), (310,842), (313,840), (314, 841), (315,653), (318,777), (320,780), (336,694), (337,862), (338,849), (339,718), (342,859), (352,804), (359,864), (363,815), (378,696), (380,851), (382,861), (391,750), (397,747), (428,594), (429,596), (432,599), (436,604), (474,626), (477,633), (502,673), (516,697), (520,707), (540,699), (557,743), (573,752).
16 centrosymmetric when open: (112,789), (113,790), (120,487), (124,306), (175,803), (176,808), (203,562), (206,395), (289,834), (290,835), (297,635), (360,850), (361,860), (392,746), (476,628), (558,741).
60 axially symmetric when closed: (9,410), (11,412), (22,214), (28,422), (33,578), (35,576), (43,686), (63,679), (67,512), (90,662), (96,329), (99,327), (145,732), (136,722), (161,403), (162,712), (177,372), (178,536), (180,551), (195,569), (196,402), (197,563), (201,748), (204,744), (229,426), (231,424), (234,584), (239,241), (243,606), (245,608), (260,263), (265,675), (270,506), (278,495), (324,465), (346,369), (347,371), (348,567), (349,701), (350,549), (375,702), (386,527), (393,565), (396,560), (401,731), (405,728), (414,416), (431,434), (446,592), (461,658), (500,501), (517,518), (529,564), (537,704), (538,552), (566,730), (598,601), (616,618), (703,738), (714,749).
16 centrosymmetric when closed: (24,216), (30,419), (32,577), (34,575), (64,681), (68,513), (92,663), (97,330), (228,425), (230,423), (236,581), (266,676), (273,507), (325,466), (443,589), (462,659).
and 44 asymmetric pairs: (3;209), (20;212), (23;215), (26;417), (29;418), (58;248), (59;249), (128;353), (129;541), (170;357), (172;358), (181;374), (200;390), (207;555), (232;579), (235;580), (354;726), (408;736), (440;586), (441;57), (449;613), (484;643), (542;727), (571;737), (759;762), (765;782), (767;799), (770;773), (775;783), (781;805), (794;811), (796;812), (820;823), (829;831), (845;858) (847;863), (865;866), (867;868), (869;875), (870;878), (871;874), (872;877), (873;879), (876;880).
Finally there are 64 onefold magic, all axially symmetric open tours: 1, 2, 4, 5, 6, 7, 36, 37, 39, 47, 72, 75, 79, 134, 138, 139, 144, 153, 166, 167, 168, 179, 186, 194, 198, 199, 210, 211, 254, 376, 381, 404, 447, 448, 458, 480, 482, 556, 609, 610, 611, 612, 614, 619, 625, 629, 631, 641, 645, 656, 672, 680, 683, 740, 742, 745, 825, 826, 830, 832 846, 848, 856, 857. Of these 619 is another Queen tour but using four types of move, WFDT. Diagrammed is 176 which is a "semi-natural" magic square in which half the numbers occur in natural order and half in reveres order, this uses 5 move patterns.
4×4. Wizard. Diamagic tours with five move types WFNCZ. These are all the possible moves of a Wizard on this board using all its powers. Which are of this type I do not seem to have checked yet.