# Representation of Knight's Tours

© by George Jelliss, 4 January 2001, revised 12 March 2011

This article was originally the first part of a note on "Literary Tours". The second part is now under the heading Cryptotours.

Puzzle questions marked have corresponding answers marked in the Solutions section, including tour diagrams.

It seems that no published knight's tours were shown by means of geometrical diagrams until dalla Volpe in 1766, though there are signs of symmetry in some which suggest that diagrams must have been used in their construction. At least four different methods of presentation of tours using letters occur in the mediaeval manuscripts: (1) Numbering the cells, using the methods of numeration then in use, based on the successive letters of the alphabet. (2) Listing the coordinates of the cells, again using letters as coordinates rather than specialised numerals. The 'algebraic' system of notation now used by most chess players preserves this in part, using letters for files and numerals for ranks. (3) Using the literal coordinates of the cells as the first syllables in an acrostic poem. (4) The 'verse tour' method of Rudrata, as described by Murray, and termed here 'paradromic verse'.

## «Coordinates

The use of coordinates to record positions and moves in chess goes back at least to the time of al Adli (c.840). H. J. R. Murray (British Chess Magazine 1902) wrote: “The ... manuscripts of the chess work which was compiled from the books of al-Adli and as-Suli contain diagrams which represent knight's tours. In one of these ... the successive squares to which the knight is moved are marked by the old Arabic letter-numerals ... The next diagram in the manuscripts is at first sight quite enigmatic. The squares are marked with a pair of letters .... the bottom row, reading from right to left, being ya, ka, la, ma, na, ra, sha, ta; while the second row ends in b, and the other six in j, d, h, w, z and hh respectively, the initial letter of every file being the same throughout. It is obvious that this is ... only a means of describing the squares in a convenient manner ...”

The Manasollasa of Somesvara (c.1150, described by F. Bernhauer 1996) uses a similar system, the files lettered c, g, n, d, t, r, s, p and the ranks (top to bottom) by vowels similar to a, â, i, î, u, û, e, ê. This system makes it possible to present a tour as a sequence of syllables that is pronounceable and looks like Sanskrit, but (as far as I am aware) makes no sense. The tour (in the reentrant version), grouped into four-cell sections, runs:

 pa si pu se - tê ne cê gû - nî cu gi ca - nâ ta sâ pî sû pê re dê - ge dû gu ci - ga dâ ra pâ - sî pû sê te nê ce nû gê - cû gî câ na - tâ sa pi su - pe rê de nu tû rî di tu - ri dî ru ti - du rû tî ni - cî gâ da râ

An Anglo-Norman chess manuscript (~1275-1300) in the King's Library of the British Museum (according to H. J. R. Murray A History of Chess 1913) uses a,b,c,d,e,f,g,h for the files and i,k,l,m,n,o,p,q for the ranks (from top to bottom). The currently customary coordinate system used in most chess books substitutes 1,2,3,4,5,6,7,8 for the ranks (bottom to top). This 'algebraic' notation was introduced by Philip Stamma in his Essai sur le Jeu des Echecs (1737). This notation can be extended to boards of any size, with files lettered a,b,c ... and ranks numbered 1,2,3... from the bottom left corner. When we wish to apply mathematical methods however it is necessary to make both coordinates numerical, the method of Descartes (1637), the cell in file x and rank y being denoted (x, y).

A similar consonant-vowel system of coordinates is used in the telegraphic code for transmission of chess games, known as the Gringmuth notation in which the files a–h are lettered BCDFGHKL on White's half of the board and MNPRSTWZ on Black's side, while the ranks 1–4 and 8–5 are lettered AEIO, so that each cell has a two-letter designation. For example castling king-side with the White king is shown as GAKA and with the Black king as SAWA. Beverley's tour in this notation, divided into four-cell sections again, is almost an incantation:

 MA PE NO RI - SA WE ZO TI - GO KI LA HE - FA CE DO BI DE BA CI FO - GE KA LI HO - SI WO ZE TA - RE NA MI PO BO DI CA FE - HA LE KO GI - TO ZI WA SE - RO NI PA ME PI MO NE RA - TE ZA WI SO - HI LO KE GA - FI CO BE DA

One can of course easily 'demonstrate' the knight's tour just by memorising a simple open tour. An elaborate verbal scheme for memorising a tour was described by George Walker in an article on 'Chess without a Chessboard' in Fraser's Magazine (March 1840). According to Roget (1840) this described a method “designating each square by a different syllable composed of certain consonants and vowels, indicating the horizontal and vertical columns in which it stands. The whole series of these 64 arbitrary syllables, joined into 16 words, to be learned by heart.” I have not seen the original reference, but the method seems to be that described by William Mason in the Good Companions Chess Problem Club Folders (1917 pp.207-8): “A stunt I used to do over fifty years ago — moving the knight to each square of the board, with my eyes blindfolded or my back turned to it.” The ranks are named: un, oo, ee, or, iv, ix, en, et and the files: M, L, K, H, G, F, D, B. This provides monosyllabic names for the 64 squares, which can then be memorised, in the order of the tour, as a piece of nonsense doggerel.

The tour is represented as follows:

 Len Het Fen Bet Dix Bor Doo Gun Koo Mun Lee Kun Moo Kee Goo Dun Bee Div Ben Fet Hen Let Mix Lor Hee Kiv Gor Hix Liv Men Ket Gen Kix Giv Fee Hor Gix For Hiv Gee Fiv Den Biv Dee Bun Foo Hun Loo Mor Lix Met Ken Get Fix Det Bix Dor Boo Fun Hoo Lun Mee Kor Miv

## «Acrostics

Some of the tours in the Arabic manuscripts were presented by means of an acrostic verse, the first two letters of the 64 lines of verse giving the coordinates of the cells. However, there are no modern examples of this type of acrostic composition because of the modern use of semi-numerical coordinates. Perhaps some poet will undertake the task of completing an appropriate set of 16 quatrains for the Beverley tour, or some other, using the letter-coordinates of the Gringmuth notation, as listed above, as the initial pairs of letters!

The following 'kooky gibberish' is the best I have managed so far. The coordinates are the first two letters of the words:

Magical Pegasus Nobly Ride, Sagely Wend Zonal Tiles,
Go Kind Labyrinthine Hero, Fabulous Centaurs Doubly Bind,
Delineate Baffling Circles Fourfold, Generate Kaleidoscopic Lively Horsemanship,
Sixty-four Worlds Zealously Take, Rectangulate Nature's Mighty Power,
Boldly Direct Careful Feet, Harmoniously Leap Kooky Gibberish,
Tour Zigzag Wander Serpentine, Rove Nightmarish Palamedean Meanders,
Pirouetting Motion Never Rash, Teach Zany Wisdom So,
High Low Keener Grow, Finally Completing Beverley's Dance.

Rudrata (900ad) apparently presented his tours by syllables on the cells which when read normally or in the sequence of the tour give the same verse. By my calculations this means that the syllable patterns in his three tours, rook, elephant (two forms) and knight have to be as shown:

Murray argues that the elephant tour is split at the halfway point because of the difficulty of fitting a verse to the pattern that results if the middle move is a simple forward step (case 3 above) since this “allows the use of only two different syllables in the third and fourth lines” and “such a task approaches sufficiently near to impossibility to justify the abandonment of the chess condition in part.” However, this argument is unconvincing, since in the knight's tour apart from the 1st and 21st cells the others use only two syllables, and this was apparently achieved. A simpler explanation is the wish to retain the poetic form of two matching couplets.

It is natural to wonder whether it is possible to fit English verses to tours in the fashion described above. Not surprisingly, no other examples of this method of presentation of tours have been reported, other than the following rather absurd little rook-tour reverse-verses, and the even more absurd elephant tour couplet which I gave in Chessics 1985. Punctuation, pronunciation and spelling are open to poetic licence!

## «Alphabetical Numbering

Many of the earliest tours were presented simply by lettering the successive squares visited by the knight in alphabetical order, since at that time the symbols for numbers were not separate from those for the sounds of speech. I do not know if any knight's tours incorporated words in this way, but Singmaster (1991) reports that al Buni (~1200ad) presented some 4×4 magic squares in alphabetical form with the top rank spelling a word.

† 1: Using A to P for 1 to 16 construct a 4×4 magic square including a word in one rank.

† 2: The earliest examples I know of that apply this principle to knight's tours are the dedicatory lettered tours employed by T. R. Dawson on the title pages of four of his books in the C. M. Fox series, which began in 1935. Dawson's tours are all lettered A...Z&A...Z&A...J and are all closed. The ampersands are inserted between Z and A to give an odd number of symbols, so that a letter can occur on cells of either colour. We show his WILD ROSES example from Caissa's Wild Roses in Clusters 1937.

 Z J M X A F K H N Y & I L I B E A Y W Z C D G J V O B E H & D C X K U O S E G A P N R D F P S F J W L T H U B Q M Q I V C R G T
 B D D M & F F K E M A E G L Z G C C F L N H J E S K N I G H T Y N B T O U R H I U R J O K X X S A M P W Z V Q I Q V & L P J Y W

† 3: It gives more scope for showing difficult words or phrases if one allows the tour to be open and to start at any letter of the alphabet. Here is my tour, from Chessics 1978, that spells out KNIGHT TOUR and runs from E(ast) to N(orth).

† 4: In Chessics (#15, p.4, 1983) I used a tour lettered Q to Z spelling out CHESSAYS along the top rank to advertise a series of booklets with that title.

 C H E S S A Y S F R B B Q T R & I D G T T Z R X Q A C Z U W I Q D J U U H Y V K & P G M V J P X K E N Y V N L W O Z L F M X W O
 C D Y G W Z U Y Z H E A F W L Y C D E R F T G V I W T S H K X M U B Q J S N U A X K V O B I J Q M Z V P X P C T N O L A D B R S

† 5: A normal tour spelling out CHESSICS is impossible because of two Cs on one colour and two Ss on the other colour. Nevertheless Clive Grimstone, on the same page, managed to show CHESSICS along a diagonal by the drastic device of joining each pair of opposite sides of the board like a cylinder but with a Moebius twist – the result of which is known as an 'RP-Board' – he manages this also without employing ampersands.

These word-spelling tours of course tend to be extremely irregular, since they are constructed to fit the given letters. There is much scope for amusement in trying to get a tour to spell out a phrase of your own choosing. The initial and final letters can be any of the following pairs: AJ, BK, CL, DM, EN, FO, GP, HQ, IR, JS, KT, LU, MV, NW, OX, PY, QZ, R&, SA, TB, UC, VD, WE, XF, YG, ZH, &I. To find which letters appear on which colour it helps to write the alphabet in a zigzag form: A B C D ··· The interest in these tours is in constructing them rather than tracing the tour from the given letters, which is usually quite easy. Diagrams of the tours are given at the end for those who wish to trace them.

† 6: Alphabetical tours can also be constructed to show other tricks. The following tour (a K...T tour, from Chessics #22, p.67, 1985) depicts the 6-cell routes (lettered KNIGHT) of the two black knights in visiting their white cousins. Although the routes the two knights take may seem somewhat arbitrary, in fact they are the ONLY such routes possible under the alphabetical conditions. The Hs and Is have to cross-connect.

 C K E Y A M K W F Z B L J X K N B D N & L N V L O G A I M J O O & C R I P G M U S P H F X U P S D Z R Q H R T V Q T E Y S W T O
 U Q W F S S Y U H G T R X V R Q P V E J O E T Z H I O D W & P Q N D I N K P F A J A L X C M I O C M Z L M K B G & K B L Y H N J

† 7: One of the attractions of these alphabetical tours is the ambiguity that sometimes arises, that can send you off on the wrong track when trying to trace the tour. For example in the tour shown above left one starts b8(K)-d7(L) but then has a choice of two Ms at f8 or e5. The scheme shown on the right (also from Chessics #22 1985) is an attempt to maximise the number of such ambiguities. The tour runs from H to Q and can be done in 16 different ways, there being two choices at each of the bold-printed letters I, K, M, and O.

† 8: "Revolver Practice": Construct a tour spelling out REVOLVER on the top rank. (This was suggested by C. J. Morse who claims that this is the only word whose letters form the same pattern as the chessmen on the back row in the opening position!)

## «Solutions

Alphabetical Numbering

‡ 1: Lettered magic squares.
By reversing the second the word is altered to LOBE in the second row.
 A N K H X P F K A X F I D O G L M B X E B O L X L G N A P C F I X I N C H X M B K H J E D O X D G J M X C P E J

‡ 2-8: Diagrams of the tours:

‡ 8: Revolver practice.
 R E V O L V E R W N Q U U S K W D S F M P F Q D N X J T T J X O I C M G G P C I Y O H K T H N Y B L Q & & L S B P Z A K R A Z M