THE KNIGHT'S TOUR: A Chessboard Curiosity

Note by Robert L. Birch, based on Euler and Fauvel-Gouraud (posthumous comments by jrb ‘071105)

The exercise known as the "knight's tour" consists of moving the knight around the chessboard, by knight's moves, so as to touch each square once only. The knight’s tour is not a part of the game of chess but is performed on the usual chessboard.

Leonhard Euler (a blind Swiss mathematician who pronounced his last name "oiler") took about four years to develop the knight’s tour. In 1845 a French teacher of the art of memory, Francis Fauvel-Gouraud, taught his students how to memorize the steps of the knight’s tour. A combination of these efforts by Euler and Fauvel-Gouraud permits a careful student to learn the steps by heart.

The tour can start from any square and end one knight’s jump away from the starting point. The white king’s knight’s home square can be the starting point[jrb1] , and the final step can be to the white king’s pawn’s home square.

For this exercise, the squares of the chessboard can be numbered by column and row, so that 11 represents the white square at the top (far) left of the board (the black queen’s rook’s square) and 18 is the number of the lower left square (which is the white queen’s rook’s square).

The pattern for numbering the squares of the chessboard is as follows[jrb2] :

11	21	31	41	51	61	71	81
12	22	32	42	52	62	72	82
13	23	33	43	53	63	73	83
14	24	34	44	54	64	73	84
15	25	35	45	55	65	75	85
16	26	36	46	56	66	76	86
17	27	37	47	57	67	77	87
18	28	38	48	58	68	78	88

The memory-aid tradition taught by Fauvel-Gouraud to help his students learn the knight’s tour includes the substitution of consonants for the numbers and then the building of a story, so that the key words of the story encode the numbers of the squares to which the successive moves are made. Starting from square 78, the white king’s knight’s square, successive steps are to 86, 74, 82, 61 and so forth around the board in a complex sequence.

A story to recall the successive squares can start with naming the white king’s knight "Goofy" and the pretense that he starts a trip around a prospective battlefield. His first step is to square 86; he then goes on to 74, 82, 61, 42, 21, and 13. The squares involved can be recalled from the words goofy, fish, car, van, jet, horn, net and dome. These words are based on Fauvel-Gouraud’s listing of the consonants as used to encode the numbers. The correlation pattern is as follows:

DIGIT-CONSONANT CORRELATION PATTERN USED FOR ENCODING NUMBERS IN MNEMONICS

0=S, Z, or soft-C

1=D, T, or TH

2=N

3=M

4=R

5=L

6=J, SH, or soft-CH or the soft-G of ginger

7=K, Q, hard-CH as in chronic or hard-G as in goat

8=F or V

9=B or P

The process of learning the successive steps of the knight’s tour by the memory-aid system of Fauvel-Gouraud can be reduced to making up a story including words that encode the successive target squares. These can be placed in eight octets numbered 11-18, 21-28, 31-38, 41-48, 51-58, 61-68, 71-78. and 81-88[jrb3] .

If it is decided to start the tour from the white king’s knight’s home square (square 78), the first word of the story should encode 78 by using the sounds for 7 and 8. The words café, coffee, Goofy, or quaff can be used.

Goofy can be imagined in a café (78) from which he moves to square 86, which can be coded by fish or fudge. "Fish" can be imagined as a fish working as a psychiatrist, Dr. Fish, who lends Goofy a car. The car has the k and r sounds needed to encode 74, the next square of the tour. The fourth square, 82, can be coded by "van" or by "fan." The car can be imagined being traded for a van with a broken fan.

Thus far, the story recalls the numbers 78, 86, 74, and 82.

The next four numbers are 61, 42, 21, and 13. These can be associated with a story that Goofy leaves the van to get on a jet which blows a horn as it takes off and hits a net from which it turns and hits a dome. These key words can recall the next squares, as specified, with the sounds j-t, r-n, n-t, and d-m.

The eight octets (sixty-four steps)of the knight’s tour can be coded as follows:

*1-64*	*Step #s*	Square #
0-7	11-18	78 Goofy	86 fish	74 car	82 van	61 jet	42 runway	21 net	13 dome
8-15	21-28	25 nail	17 tag	36 match	28 knife	47 rock	66 judge	54 lure	46 arch
16-23	31-38	34 hammer	15 dial	23 gnome	11 tooth	32 moon	51 wallet	72 cane	84 fur
24-31	41-48	76 coach	88 fife	67 jockey	48 reef	27 ink	35 mill	16 dish	24 wiener
32-39	51-58	12 tine	31 mat	52 lion	71 cat	83 foam	75 igloo	87 fog	68 chef
40-47	61-68	56 leash	64 chair	43 ram	55 lily	63 gem	44 warrior	65 shell	53 loom
48-55	71-78	45 rail	33 mummy	14 tower	22 nun	41 heart	62 chain	81 foot	73 comb
56-63	81-88	85 file	77 cake	58 leaf	37 mike	18 dove	26 hinge	38 muff	57 lock

(Step octets above, not shown below.)

Starting from the white king’s knight’s square (0 of 63).

	1	2	3	4	5	6	7	8
1	19	6	33	52	21	4	35	54	1
2	32	51	20	5	34	53	22	3	2
3	7	18	49	42	47	44	55	36	3
4	50	31	16	45	14	41	2	23	4
5	17	8	29	48	43	46	37	56	5
6	30	61	10	15	40	13	24	1	6
7	9	28	59	12	63	26	57	38	7
8	60	11	62	27	58	39	0	25	8
	1	2	3	4	5	6	7	8

.____.

It might be interesting to explore the obvious and not so obvious permutations of this path.

.____.

Starting step 1 of 64 from square 11.

	1	2	3	4	5	6	7	8
1	1	14	51	32	63	16	49	30	1
2	52	33	64	15	50	31	62	17	2
3	13	2	35	42	37	40	29	48	3
4	34	53	4	39	6	43	18	61	4
5	3	12	55	36	41	38	47	28	5
6	54	23	10	5	44	7	60	19	6
7	11	56	25	8	21	58	27	46	7
8	24	9	22	57	26	45	20	59	8
	1	2	3	4	5	6	7	8

11	21	31	41	51	61	71	81
12	22	32	42	52	62	72	82
13	23	33	43	53	63	73	83
14	24	34	44	54	64	73	84
15	25	35	45	55	65	75	85
16	26	36	46	56	66	76	86
17	27	37	47	57	67	77	87
18	28	38	48	58	68	78	88

	1	2	3	4	5	6	7	8
1	19	6	33	52	21	4	35	54	1
2	32	51	20	5	34	53	22	3	2
3	7	18	49	42	47	44	55	36	3
4	50	31	16	45	14	41	2	23	4
5	17	8	29	48	43	46	37	56	5
6	30	61	10	15	40	13	24	1	6
7	9	28	59	12	63	26	57	38	7
8	60	11	62	27	58	39	0	25	8
	1	2	3	4	5	6	7	8

	1	2	3	4	5	6	7	8
1	1	14	51	32	63	16	49	30	1
2	52	33	64	15	50	31	62	17	2
3	13	2	35	42	37	40	29	48	3
4	34	53	4	39	6	43	18	61	4
5	3	12	55	36	41	38	47	28	5
6	54	23	10	5	44	7	60	19	6
7	11	56	25	8	21	58	27	46	7
8	24	9	22	57	26	45	20	59	8
	1	2	3	4	5	6	7	8

11	21	31	41	51	61	71	81
12	22	32	42	52	62	72	82
13	23	33	43	53	63	73	83
14	24	34	44	54	64	73	84
15	25	35	45	55	65	75	85
16	26	36	46	56	66	76	86
17	27	37	47	57	67	77	87
18	28	38	48	58	68	78	88

	1	2	3	4	5	6	7	8
1	19	6	33	52	21	4	35	54	1
2	32	51	20	5	34	53	22	3	2
3	7	18	49	42	47	44	55	36	3
4	50	31	16	45	14	41	2	23	4
5	17	8	29	48	43	46	37	56	5
6	30	61	10	15	40	13	24	1	6
7	9	28	59	12	63	26	57	38	7
8	60	11	62	27	58	39	0	25	8
	1	2	3	4	5	6	7	8

	1	2	3	4	5	6	7	8
1	1	14	51	32	63	16	49	30	1
2	52	33	64	15	50	31	62	17	2
3	13	2	35	42	37	40	29	48	3
4	34	53	4	39	6	43	18	61	4
5	3	12	55	36	41	38	47	28	5
6	54	23	10	5	44	7	60	19	6
7	11	56	25	8	21	58	27	46	7
8	24	9	22	57	26	45	20	59	8
	1	2	3	4	5	6	7	8

Square #

11	21	31	41	51	61	71	81
12	22	32	42	52	62	72	82
13	23	33	43	53	63	73	83
14	24	34	44	54	64	73	84
15	25	35	45	55	65	75	85
16	26	36	46	56	66	76	86
17	27	37	47	57	67	77	87
18	28	38	48	58	68	78	88

	1	2	3	4	5	6	7	8
1	19	6	33	52	21	4	35	54	1
2	32	51	20	5	34	53	22	3	2
3	7	18	49	42	47	44	55	36	3
4	50	31	16	45	14	41	2	23	4
5	17	8	29	48	43	46	37	56	5
6	30	61	10	15	40	13	24	1	6
7	9	28	59	12	63	26	57	38	7
8	60	11	62	27	58	39	0	25	8
	1	2	3	4	5	6	7	8

	1	2	3	4	5	6	7	8
1	1	14	51	32	63	16	49	30	1
2	52	33	64	15	50	31	62	17	2
3	13	2	35	42	37	40	29	48	3
4	34	53	4	39	6	43	18	61	4
5	3	12	55	36	41	38	47	28	5
6	54	23	10	5	44	7	60	19	6
7	11	56	25	8	21	58	27	46	7
8	24	9	22	57	26	45	20	59	8
	1	2	3	4	5	6	7	8