Fiveleaper Tours

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Sections on this page:
Introduction — Various Fiveleaper Tours — A Double 8×8 Tour by the Fiveleaper — Fiveleaper Pseudotours 8×8 — Magic Fiveleaper Tours 8×8: Notes — The 42 Closed Magic Tours — The 16 Open Magic Tours — End

é Introduction

Just as the knight makes moves of length root-5 that have coordinates {1,2}, a fiveleaper is a type of generalised knight that makes moves of length 5 units, with coordinates either {0,5} or {3,4}. I'm not sure when the fiveleaper was first introduced as a fairy chess piece, but T. R. Dawson gave an analysis of multipattern fixed-distance leapers, of which the fiveleaper is the simplest example, in Chess Amateur August 1925. For more details see the section on Compound Leapers on the Theory of Moves page. The leaper having only the {3,4} move is known as an Antelope and some results using it, including a tour on the 14×14 board, are given on the Longer Leapers page.

M. Kraitchik gave what is probably the first fiveleaper tour of the 8×8 board in 1927 (see diagram below). Because of difficulty in showing fiveleaper tours clearly in graphical form, they are given here as numerical arrays.

The catalogue of magic fiveleaper tours by T. W. Marlow, completed in 1990, is published in full here for the first time. This work was previously reported briefly with a few examples in The Probemist and Variant Chess in 1991.

é Various Fiveleaper Tours

T. H. Willcocks, Chessics 24 Winter 1985 p.93; Open 5-leaper tour on 7×7 less centre cell.

04 23 46 27 42 11 02

25 40 29 36 19 44 09

16 13 34 07 38 17 14

21 32 01 05 22 31

48 43 10 03 24 47 28

37 18 45 26 41 12 35

06 39 30 15 20 33 08

T. W. Marlow Chessics 24 Winter 1985 p.93; closed fiveleaper tour on 6×9 board.

54 21 14 31 18 11 28 15 52

23 50 07 46 33 04 49 08 45

36 43 26 39 02 35 42 25 38

19 12 29 16 53 20 13 30 17

32 05 48 09 22 51 06 47 10

01 34 41 24 37 44 27 40 03

Marlow wrote: “This is symmetrical about a vertical axis and can be shown to be the smallest area even for an open tour. For a shorter dimension, less than 6, there will be a row, e.g. 3, of squares which all have no more than one exit. For 6×8, or less, there are at least 4 central squares, de34, which have only one exit. This leaves only 7×7 where there is no exit from the central square.”

Maurice Kraitchik, Le Probleme du Cavalier 1927 p.74. Fiveleaper closed tour 8×8.

40 33 16 59 62 13 34 47

45 26 51 06 29 44 23 52

64 11 56 19 38 03 08 57

31 14 35 38 41 32 15 36

28 61 22 53 46 27 60 21

39 04 17 58 63 12 05 18

42 25 50 07 30 43 24 49

01 10 55 20 37 02 09 54

G. P. Jelliss, Chessics vol.1 #5 1978 p.8. Fiveleaper centro-symmetric closed tour, with 32 straight and 32 skew leaps.

60 31 34 17 08 57 30 33

19 36 53 14 47 22 37 52

10 45 26 41 50 11 44 27

07 56 03 64 61 06 55 16

48 23 38 29 32 35 24 39

59 12 43 18 09 58 13 42

20 05 54 15 46 21 04 51

01 62 25 40 49 02 63 28

é A Double 8×8 Tour by the Fiveleaper

In Variant Chess (vol. 1, isue 6, April-June 1991, page 75) I made the following observation: “Since the fiveleaper has four moves at every square of the 8×8 board it follows that in every closed tour the unused moves are also two at every square, and therefore form either a tour (is this possible?) or a pseudotour (i.e. a set of closed circuits). To use network-theory terminology, this would be a pair of Hamiltonian tours that together form an Eulerian tour. A trivial example of this is provided by the moves of a wazir on a 2×2 torus.” The term ‘Eulerian tour’ is used here in the sense of a path that uses every branch of a network once.

The question of whether such a double tour is possible was in fact answered in the affirmative by Tom Marlow in a letter to me of 17 November 1991, but due to an oversight his result was not published until ten years later, in the last issue of The Games and Puzzles Journal (vol.2, issue 18, March 2001, page 347). The following is Marlow's solution; in his own words: “The 5-leaper has exactly four moves available on every square of the 8×8 board. In all there are 128 leaps, each being possible from either end. The two closed tours below make use between them of all these leaps. The method of construction was to build a tour starting at a1 and at each leap to mark as unavailable the corresponding leap after 180 degree rotation; e.g. the opening a1-a6 barred h8-h3 and h3-h8. When the tour was complete the same route, rotated 180 degrees, could be travelled using the barred leaps. That tour was then renumbered to start at a1.”

5-leaper double tour, part 1

20 47 62 55 06 21 46 63

31 42 57 50 11 34 29 44

02 59 16 09 52 03 36 17

13 22 39 26 19 14 23 38

54 05 28 45 64 61 56 07

51 48 35 30 43 58 49 10

32 41 24 37 12 33 40 25

01 60 15 08 53 04 27 18

5-leaper double tour, part 2

46 37 60 11 56 49 04 63

39 24 31 52 27 40 23 32

54 15 06 21 34 29 16 13

57 08 03 64 19 36 59 10

26 41 50 45 38 25 42 51

47 28 61 12 55 48 05 62

20 35 30 53 14 07 22 33

01 18 43 58 09 02 17 44

é Fiveleaper Pseudotours 8×8

The following results are by Tom Marlow, October 1990. They enumerate the ways of forming two or four paths that by reflection or rotation can be placed to cover the 8×8 board, using each cell once, as in Vandermonde's method for the knight. The following is Marlow's own text:

Chains of 16 squares that reflect or rotate to cover the whole board.

All chains start at square 1. Two-fold chains end one move from square 64 and so can then be repeated after rotation by 180 degrees to make closed loops of 32 squares. Four-fold chains end one move from square 1, so form closed loops. Squares 36 and 29 are one move from both corner squares, i.e. 1 and 64. Consequently chains that end on either can be used in two-fold or four-fold form.

Reflective loops can then be reflected about a vertical axis (and a horizontal axis of four-fold) to cover the whole board. Similarly rotatory loops can be turned, 180 degrees if two-fold, or in three steps of 90 degrees if four-fold, to cover the whole board.

Two-fold reflective (3 cases):
01, 41, 46, 11, 39, 04, 32, 52, 23, 63, 28, 56, 21, 50, 30, 59
01, 41, 46, 18, 53, 25, 60, 20, 55, 27, 07, 36, 16, 11, 39, 59
01, 41, 21, 50, 30, 02, 37, 09, 14, 43, 23, 52, 32, 04, 39, 59
Four-fold reflective (3 cases):
01, 41, 46, 11, 39, 04, 33, 13, 42, 02, 37, 09, 44, 15, 35, 06
01, 41, 46, 18, 53, 25, 05, 45, 10, 38, 58, 29, 49, 54, 26, 06
01, 41, 21, 50, 30, 02, 37, 09, 14, 43, 23, 52, 32, 51, 26, 06
Two-fold and Four-fold reflective (12 cases):
01, 41, 46, 06, 35, 15, 44, 09, 14, 34, 05, 40, 12, 47, 07, 36
01, 41, 46, 06, 35, 15, 44, 09, 14, 34, 05, 25, 53, 18, 58, 29
01, 41, 46, 11, 16, 45, 10, 38, 03, 31, 60, 40, 12, 47, 07, 36
01, 41, 46, 11, 16, 45, 10, 38, 03, 31, 60, 25, 53, 18, 58, 29
01, 41, 21, 50, 30, 02, 42, 13, 33, 61, 26, 06, 46, 11, 16, 36
01, 41, 21, 50, 30, 02, 42, 13, 33, 04, 39, 59, 19, 54, 49, 29
01, 41, 12, 40, 05, 45, 10, 38, 58, 18, 46, 06, 26, 54, 49, 29
01, 41, 12, 40, 05, 34, 62, 27, 55, 20, 49, 54, 19, 47, 07, 36
01, 41, 12, 40, 05, 34, 14, 09, 44, 15, 35, 06, 46, 18, 58, 29
01, 41, 12, 40, 60, 31, 03, 38, 10, 45, 16, 11, 46, 18, 58, 29
01, 41, 12, 40, 60, 31, 51, 56, 21, 50, 30, 59, 19, 47, 07, 36
01, 41, 12, 40, 60, 20, 55, 27, 07, 47, 19, 59, 39, 11, 16, 36
Two-fold reflective and rotatory (1 case):
01, 41, 46, 11, 16, 36, 07, 47, 12, 40, 35, 15, 44, 04, 39, 59
Two-fold and Four-fold rotatory (3 cases):
01, 41, 46, 06, 26, 61, 21, 50, 30, 25, 53, 18, 23, 63, 58, 29
01, 41, 12, 40, 35, 15, 44, 04, 39, 59, 19, 54, 14, 09, 49, 29
01, 41, 12, 40, 35, 15, 44, 04, 39, 59, 19, 47, 42, 02, 07, 36

é Magic Fiveleaper Tours 8×8

Notes

The following text and results are by Tom Marlow, September 1990. (It seems surprising that there are no magic tours with the initial cell on the first rank. Can anyone put forward an explanation of why this is so?)

The diagrams show 58 five-leaper magic tours. All are magic in the sense that all ranks and files sum to the magic constant of 260. Numbers #5 and #47 [red diagrams] are fully magic because additionally their diagonals have the same sum. (These two were published in The Probemist March 1991, and the other results, with two examples, were reported in Variant Chess, issue 6, April-June 1991, p.75.)

The method of construction is to find sequences of 32 five-leaper steps that, when reflected in the horizontal axis, cover the remaining 32 squares of the board. Then if the 32nd square is on the second or seventh rank its reflection is five squares away and the second half of the tour proceeds in reverse order to the first half. [The cells containing the numbers 1, 16, 17, 32, 33, 48, 49, 64 are highlighted.] The result is that all vertical columns sum to 260 because each consists of four reflecting pairs such as (64,1) or (60,5) which each sum to 65. It remains to find cases where the horizontal rows also total 260.

42 of the tours begin on the second rank so are closed, i.e. the end is one five-leaper move from the start. Consequently they can be renumbered from 1 to 64 in the sequence 32, 31, ..., 1, 64, ..., 33 and in most cases remain magic. 34 cases, marked («) [blue diagrams] are unchanged by this transformation because the 32 square sequence is symmetric about the vertical axis. Of the rest, 7 marked (*) remain magic. The exception is number #20. Number #5 which is fully magic is in the (*) category and remains fully magic under the transformation.

Furthermore, tours #12 and #15 [mauve diagrams] can be renumbered from 1 in the sequence 49, ..., 64, 1, ..., 48 when they remain magic and have the («) category.

é The 42 Closed Magic Tours

#1 *
11	52	21	42	07	60	29	38
64	19	16	09	40	33	62	17
55	14	31	02	47	50	35	26
24	59	04	37	12	45	22	57
41	06	61	28	53	20	43	08
10	51	34	63	18	15	30	39
01	46	49	56	25	32	03	48
54	13	44	23	58	05	36	27

#2 «
35	48	25	12	21	08	49	62
64	27	10	51	46	23	06	33
29	18	41	60	37	56	15	04
20	43	58	31	02	39	54	13
45	22	07	34	63	26	11	52
36	47	24	05	28	09	50	61
01	38	55	14	19	42	59	32
30	17	40	53	44	57	16	03

#3 *
19	42	61	14	35	20	07	62
64	25	16	37	12	33	56	17
27	22	05	10	39	44	59	54
52	31	08	47	02	41	50	23
13	34	57	18	63	24	15	36
38	43	04	55	26	21	06	11
01	40	49	28	53	32	09	48
46	23	04	51	30	45	58	03

#4 *
13	62	35	20	09	24	61	36
64	33	22	15	58	39	18	11
51	06	27	46	55	42	05	28
08	25	48	53	02	31	44	49
57	40	17	12	63	34	21	16
14	59	38	19	10	23	60	37
01	32	43	50	07	26	47	54
52	03	30	45	56	41	04	29

#5 * diagonals magic
03	38	59	22	11	30	37	60
64	25	08	45	52	33	24	09
19	48	15	42	55	18	49	14
12	31	36	61	02	39	58	21
53	34	29	04	63	26	07	44
46	17	50	23	10	47	16	51
01	40	57	20	13	32	41	56
62	27	06	43	54	35	28	05

#6 *
03	38	51	58	23	30	11	46
64	25	16	05	44	33	56	17
61	28	13	08	41	36	53	20
22	31	10	47	02	39	50	59
43	34	55	18	63	26	15	06
04	37	52	57	24	29	12	45
01	40	49	60	21	32	09	48
62	27	14	07	42	35	54	19

#7 «
51	48	29	08	25	04	49	46
64	27	34	23	10	63	06	33
13	54	21	58	39	12	43	20
56	03	60	15	18	37	30	41
09	62	05	50	47	28	35	24
52	11	44	07	26	53	22	45
01	38	31	42	55	02	59	32
14	17	36	57	40	61	16	19

#8 «
57	38	29	14	19	04	59	40
64	17	34	21	12	63	16	33
09	54	23	50	47	10	43	24
52	03	60	07	26	37	30	45
13	62	05	58	39	28	35	20
56	11	42	15	18	55	22	41
01	48	31	44	53	02	49	32
08	27	36	51	46	61	06	25

#9 «
19	48	13	36	61	20	49	14
64	11	34	27	06	63	22	33
09	58	25	42	55	08	39	24
60	03	44	15	18	53	30	37
05	62	21	50	47	12	35	28
56	07	40	23	10	57	26	41
01	54	31	38	59	02	43	32
46	17	52	29	04	45	16	51

#10 «
23	40	11	52	45	22	57	10
64	27	34	15	18	63	06	33
29	48	03	60	37	30	49	04
46	21	58	09	24	39	12	51
19	44	07	56	41	26	53	14
36	17	62	05	28	35	16	61
01	38	31	50	47	02	59	32
42	25	54	13	20	43	08	55

#11 «
24	17	46	35	62	51	16	09
11	64	21	60	37	12	33	22
40	27	58	31	02	39	06	57
29	52	15	42	55	18	45	04
36	13	50	23	10	47	20	61
25	38	07	34	63	26	59	08
54	01	44	05	28	53	32	43
41	48	19	30	03	14	49	56

#12 «
04	43	24	47	50	09	54	29
27	64	07	52	45	26	33	06
62	21	40	31	02	57	12	35
19	16	55	60	37	42	17	14
46	49	10	05	28	23	48	51
03	44	25	34	63	08	53	30
38	01	58	13	20	39	32	59
61	22	41	18	15	56	11	36

#13 «
04	49	14	53	44	19	48	29
27	64	07	42	55	26	33	06
62	09	40	31	02	57	24	35
11	20	47	60	37	50	13	22
54	45	18	05	28	15	52	43
03	56	25	34	63	08	41	30
38	01	58	23	10	39	32	59
61	16	51	12	21	46	17	36

#14 «
52	17	46	25	08	51	16	45
11	64	23	38	59	10	33	22
62	05	36	31	02	61	28	35
07	56	15	44	53	18	41	26
58	09	50	21	12	47	24	39
03	60	29	34	63	04	37	30
54	01	42	27	06	55	32	43
13	48	19	40	57	14	49	20

#15 «
08	43	24	47	50	09	54	25
27	64	05	52	45	28	33	06
62	21	36	31	02	61	12	35
19	16	55	58	39	42	17	14
46	49	10	07	26	23	48	51
03	44	29	34	63	04	53	30
38	01	60	13	20	37	32	59
57	22	41	18	15	56	11	40

#16 «
40	27	62	21	12	35	06	57
19	64	15	42	55	18	33	14
24	37	04	45	52	29	60	09
11	48	31	58	39	02	49	22
54	17	34	07	26	63	16	43
41	28	61	20	13	36	05	56
46	01	50	23	10	47	32	51
25	38	03	44	53	30	59	08

#17 «
46	17	62	05	28	35	16	51
11	64	23	58	39	10	33	22
20	25	56	53	44	41	08	13
27	36	31	50	47	02	61	06
38	29	34	15	18	63	04	59
45	40	09	12	21	24	57	52
54	01	42	07	26	55	32	43
19	48	03	60	37	30	49	14

#18 «
36	27	62	21	12	35	06	61
19	64	15	42	55	18	33	14
24	39	08	45	52	25	58	09
11	48	31	60	37	02	49	22
54	17	34	05	28	63	16	43
41	26	57	20	13	40	07	56
46	01	50	23	10	47	32	51
29	38	03	44	53	30	59	04

#19 «
44	17	52	07	26	45	16	53
23	64	09	36	61	24	33	10
28	47	14	59	38	19	50	05
03	40	31	54	43	02	57	30
62	25	34	11	22	63	08	35
37	18	51	06	27	46	15	60
42	01	56	29	04	41	32	55
21	48	13	58	39	20	49	12

#20
36	03	42	53	46	35	04	41
27	64	55	14	17	44	33	06
08	49	22	39	58	09	50	25
47	20	31	60	37	02	11	52
18	45	34	05	28	63	54	13
57	16	43	26	07	56	15	40
38	01	10	51	48	21	32	59
29	62	23	12	19	30	61	24

#21 «
60	03	46	09	24	51	30	37
11	64	21	58	39	12	33	22
06	49	18	55	42	15	48	27
25	52	31	36	61	02	45	08
40	13	34	29	04	63	20	57
59	16	47	10	23	50	17	38
54	01	44	07	26	53	32	43
05	62	19	56	41	14	35	28

#22 «
60	03	38	25	08	59	30	37
13	64	19	42	55	14	33	20
22	17	50	53	44	47	16	11
09	58	31	36	61	02	39	24
56	07	34	29	04	63	26	41
43	48	15	12	21	18	49	54
52	01	46	23	10	51	32	45
05	62	27	40	57	06	35	28

#23 «
60	03	38	11	22	59	30	37
09	64	25	52	45	08	33	24
14	49	18	55	42	15	48	19
21	58	31	36	61	02	39	12
44	07	34	29	04	63	26	53
51	16	47	10	23	50	17	46
56	01	40	13	20	57	32	41
05	62	27	54	43	06	35	28

#24 «
46	03	50	11	22	47	30	51
05	64	27	56	41	06	33	28
20	25	58	53	44	39	08	13
23	48	31	36	61	02	49	10
42	17	34	29	04	63	16	55
45	40	07	12	21	26	57	52
60	01	38	09	24	59	32	37
19	62	15	54	43	18	35	14

#25 «
54	03	44	07	26	53	30	43
05	64	19	56	41	14	33	28
10	17	50	59	38	47	16	23
25	52	31	36	61	02	45	08
40	13	34	29	04	63	20	57
55	48	15	06	27	18	49	42
60	01	46	09	24	51	32	37
11	62	21	58	39	12	35	22

#26 *
46	03	30	07	50	43	54	27
25	64	33	52	05	24	09	48
20	61	36	57	16	21	12	37
59	42	55	18	39	02	31	14
06	23	10	47	26	63	34	51
45	04	29	08	49	44	53	28
40	01	32	13	60	41	56	17
19	62	35	58	15	22	11	38

#27 «
53	04	59	08	25	38	29	44
14	17	64	55	42	33	16	19
11	62	31	50	47	02	35	22
24	39	28	45	52	05	58	09
41	26	37	20	13	60	07	56
54	03	34	15	18	63	30	43
51	48	01	10	23	32	49	46
12	61	06	57	40	27	36	21

#28 «
09	46	25	38	59	08	51	24
44	17	64	29	04	33	16	53
55	62	31	22	11	02	35	42
60	07	50	13	20	47	26	37
05	58	15	52	45	18	39	28
10	03	34	43	54	63	30	23
21	48	01	36	61	32	49	12
56	19	40	27	06	57	14	41

#29 «
11	58	23	62	35	10	39	22
60	17	64	13	20	33	16	37
53	08	41	04	29	56	25	44
46	31	50	27	06	47	02	51
19	34	15	38	59	18	63	14
12	57	24	61	36	09	40	21
05	48	01	52	45	32	49	28
54	07	42	03	30	55	26	43

#30 «
23	46	11	62	35	22	51	10
40	17	64	05	28	33	16	57
59	20	53	26	07	44	13	38
36	31	50	09	24	47	02	61
29	34	15	56	41	18	63	04
06	45	12	39	58	21	52	27
25	48	01	60	37	32	49	08
42	19	54	03	30	43	14	55

#31 «
23	46	11	62	35	22	51	10
08	17	64	37	60	33	16	25
27	20	53	58	39	44	13	06
04	31	50	41	56	47	02	29
61	34	15	24	09	18	63	36
38	45	12	07	26	21	52	59
57	48	01	28	05	32	49	40
42	19	54	03	30	43	14	55

#32 «
29	46	03	40	57	30	51	04
38	17	64	23	10	33	16	59
43	20	53	26	07	44	13	54
56	31	50	05	28	47	02	41
09	34	15	60	37	18	63	24
22	45	12	39	58	21	52	11
27	48	01	42	55	32	49	06
36	19	62	25	08	35	14	61

#33 *
13	58	03	30	47	12	59	38
50	09	64	33	20	49	10	25
23	44	61	36	41	22	05	28
46	17	54	39	14	57	02	31
19	48	11	26	51	08	63	34
42	21	04	29	24	43	60	37
15	56	01	32	45	16	55	40
52	07	62	35	18	53	06	27

#34 «
29	48	05	58	39	28	49	04
34	19	64	09	24	33	14	63
21	42	11	52	45	22	55	12
40	27	50	03	30	47	06	57
25	38	15	62	35	18	59	08
44	23	54	13	20	43	10	53
31	46	01	56	41	32	51	02
36	17	60	07	26	37	16	61

#35 «
11	56	27	36	61	06	41	22
34	17	64	13	20	33	16	63
53	08	39	30	03	58	25	44
46	05	50	23	10	47	28	51
19	60	15	42	55	18	37	14
12	57	26	35	62	07	40	21
31	48	01	52	45	32	49	02
54	09	38	29	04	59	24	43

#36 «
07	56	27	36	61	06	41	26
34	17	64	13	20	33	16	63
53	10	43	30	03	54	23	44
46	05	50	25	08	47	28	51
19	60	15	40	57	18	37	14
12	55	22	35	62	11	42	21
31	48	01	52	45	32	49	02
58	09	38	29	04	59	24	39

#37 «
06	19	54	35	62	43	14	27
25	48	09	64	33	24	49	08
60	45	12	29	04	21	52	37
31	42	15	58	39	18	55	02
34	23	50	07	26	47	10	63
05	20	53	36	61	44	13	28
40	17	56	01	32	41	16	57
59	46	11	30	03	22	51	38

#38 «
20	39	28	35	62	05	58	13
11	48	23	64	33	10	49	22
46	25	56	15	18	41	08	51
31	04	59	44	53	38	29	02
34	61	06	21	12	27	36	63
19	40	09	50	47	24	57	14
54	17	42	01	32	55	16	43
45	26	37	30	03	60	07	52

#39 «
10	49	60	03	30	37	48	23
21	58	13	64	33	20	39	12
56	15	46	25	08	51	18	41
31	36	27	54	43	06	61	02
34	29	38	11	22	59	04	63
09	50	19	40	57	14	47	24
44	07	52	01	32	45	26	53
55	16	05	62	35	28	17	42

#40 «
38	19	54	03	30	43	14	59
25	48	09	64	33	24	49	08
28	45	12	61	36	21	52	05
31	42	15	58	39	18	55	02
34	23	50	07	26	47	10	63
37	20	53	04	29	44	13	60
40	17	56	01	32	41	16	57
27	46	11	62	35	22	51	06

#41 «
58	05	40	13	20	57	28	39
11	48	23	64	33	10	49	22
02	61	30	53	44	03	36	31
19	56	15	38	59	18	41	14
46	09	50	27	06	47	24	51
63	04	35	12	21	62	29	34
54	17	42	01	32	55	16	43
07	60	25	52	45	08	37	26

#42 «
62	05	40	13	20	57	28	35
11	48	23	64	33	10	49	22
02	59	26	53	44	07	38	31
19	56	15	36	61	18	41	14
46	09	50	29	04	47	24	51
63	06	39	12	21	58	27	34
54	17	42	01	32	55	16	43
03	60	25	52	45	08	37	30

é The 16 Open Magic Tours

#43
02	39	58	51	30	23	10	47
45	28	05	32	49	44	53	04
64	25	08	13	36	41	56	17
15	22	11	62	19	38	59	34
50	43	54	03	46	27	06	31
01	40	57	52	29	24	09	48
20	37	60	33	16	21	12	61
63	26	07	14	35	42	55	18

#44
14	53	58	19	38	15	04	59
41	30	17	02	55	40	43	32
64	11	08	23	34	49	62	09
37	26	21	60	13	36	47	20
28	39	44	05	52	29	18	45
01	54	57	42	31	16	03	56
24	35	48	63	10	25	22	33
51	12	07	46	27	50	61	06

#45
56	05	24	35	14	57	44	25
11	62	37	32	49	12	19	38
64	59	42	47	02	07	22	17
15	52	45	26	55	04	29	34
50	13	20	39	10	61	36	31
01	06	23	18	63	58	43	48
54	03	28	33	16	53	46	27
09	60	41	30	51	08	21	40

#46
52	29	08	41	04	51	30	45
47	06	55	26	43	02	49	32
12	37	64	17	60	11	38	21
23	62	15	34	19	58	09	40
42	03	50	31	46	07	56	25
53	28	01	48	05	54	27	44
18	59	10	39	22	63	16	33
13	36	57	24	61	14	35	20

#47 diagonals magic
42	21	52	11	26	41	16	51
31	46	05	56	37	32	47	06
08	03	64	35	58	61	02	29
27	40	17	50	43	20	53	10
38	25	48	15	22	45	12	55
57	62	01	30	07	04	63	36
34	19	60	09	28	33	18	59
23	44	13	54	39	24	49	14

#48
36	21	58	25	12	35	16	57
23	46	05	38	55	32	47	14
28	03	64	41	52	61	02	09
11	34	17	50	43	20	59	26
54	31	48	15	22	45	06	39
37	62	01	24	13	04	63	56
42	19	60	27	10	33	18	51
29	44	07	40	53	30	49	08

#49
59	14	27	02	47	54	35	22
08	49	40	61	20	09	32	41
05	52	37	64	17	12	29	44

04	23	46	27	42	11	02
25	40	29	36	19	44	09
16	13	34	07	38	17	14
21	32	01		05	22	31
48	43	10	03	24	47	28
37	18	45	26	41	12	35
06	39	30	15	20	33	08

54	21	14	31	18	11	28	15	52
23	50	07	46	33	04	49	08	45
36	43	26	39	02	35	42	25	38
19	12	29	16	53	20	13	30	17
32	05	48	09	22	51	06	47	10
01	34	41	24	37	44	27	40	03

40	33	16	59	62	13	34	47
45	26	51	06	29	44	23	52
64	11	56	19	38	03	08	57
31	14	35	38	41	32	15	36
28	61	22	53	46	27	60	21
39	04	17	58	63	12	05	18
42	25	50	07	30	43	24	49
01	10	55	20	37	02	09	54

60	31	34	17	08	57	30	33
19	36	53	14	47	22	37	52
10	45	26	41	50	11	44	27
07	56	03	64	61	06	55	16
48	23	38	29	32	35	24	39
59	12	43	18	09	58	13	42
20	05	54	15	46	21	04	51
01	62	25	40	49	02	63	28

04	23	46	27	42	11	02
25	40	29	36	19	44	09
16	13	34	07	38	17	14
21	32	01		05	22	31
48	43	10	03	24	47	28
37	18	45	26	41	12	35
06	39	30	15	20	33	08

54	21	14	31	18	11	28	15	52
23	50	07	46	33	04	49	08	45
36	43	26	39	02	35	42	25	38
19	12	29	16	53	20	13	30	17
32	05	48	09	22	51	06	47	10
01	34	41	24	37	44	27	40	03

40	33	16	59	62	13	34	47
45	26	51	06	29	44	23	52
64	11	56	19	38	03	08	57
31	14	35	38	41	32	15	36
28	61	22	53	46	27	60	21
39	04	17	58	63	12	05	18
42	25	50	07	30	43	24	49
01	10	55	20	37	02	09	54

60	31	34	17	08	57	30	33
19	36	53	14	47	22	37	52
10	45	26	41	50	11	44	27
07	56	03	64	61	06	55	16
48	23	38	29	32	35	24	39
59	12	43	18	09	58	13	42
20	05	54	15	46	21	04	51
01	62	25	40	49	02	63	28

04	23	46	27	42	11	02
25	40	29	36	19	44	09
16	13	34	07	38	17	14
21	32	01		05	22	31
48	43	10	03	24	47	28
37	18	45	26	41	12	35
06	39	30	15	20	33	08

54	21	14	31	18	11	28	15	52
23	50	07	46	33	04	49	08	45
36	43	26	39	02	35	42	25	38
19	12	29	16	53	20	13	30	17
32	05	48	09	22	51	06	47	10
01	34	41	24	37	44	27	40	03

40	33	16	59	62	13	34	47
45	26	51	06	29	44	23	52
64	11	56	19	38	03	08	57
31	14	35	38	41	32	15	36
28	61	22	53	46	27	60	21
39	04	17	58	63	12	05	18
42	25	50	07	30	43	24	49
01	10	55	20	37	02	09	54

60	31	34	17	08	57	30	33
19	36	53	14	47	22	37	52
10	45	26	41	50	11	44	27
07	56	03	64	61	06	55	16
48	23	38	29	32	35	24	39
59	12	43	18	09	58	13	42
20	05	54	15	46	21	04	51
01	62	25	40	49	02	63	28