ç Knight's Tour Notes Index

Cryptotours

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

« Polygraphy

In the note on "Representation of Knight's Tours" the method of lettering the successive cells visited by the knight in alphabetical order is described, and examples given where this is converted into a recreation requiring the construction of a tour to show particular words spelt out by the letters in given positions on the board. A related recreation is to enter the letters along the knight's path in a sequence that spells out a sentence or a longer passage. If either the tour or the sequence of the letters is forgotten or unspecified this turns the tour into a puzzle, which French writers have termed Polygraphie.

† 1: This example is given in Staunton's chess column in the Illustrated London News (1871): “Knight's Tour No. V, by an old problem composer, respectfully dedicated to Mr Staunton. The letters, taken continuously in the order of the knight's route over the board, form a descriptive sentence which is 'Bagman's (or Piper's) News' to a chessplayer.” I think a stronger clue is necessary and will reveal that the statement is a definition of the knight's move.

S T A U N T O N R A I O N O A R H O O G Q E E M R E E T P T E X N E P C L S O C V E K O H O I N T I O S E F T R S E H G M F R S

E Y L E A N A T T R T I U S N U E T D E L E M U U I L B A L L D I B U Y U A T A P E E L R N I A R V I E B Q L E D L A E A T D S

A complete tour presented in this form is very difficult to decode without some hint as to the text. Hoffmann (1893) gives some partial tour examples using well known proverbs. A series of messages encoded letter by letter on boards of various sizes and shapes appeared in L'Echiquier in 1926–9, mainly of a salutatory nature.

† 2: An open tour in Cahiers de L'Echiquier 1930, has the title 'Polygraphie du Cavalier', from which I have taken the title of this section. This tour is shown above right. It quotes a line from Barbey d'Aurevilly, whose name appears at the end of the quote.

† 3: G. E. McGuffey, Fairy Chess Review 1937, used a quote from Ben Jonson. No other clue was given and unfortunately no-one was able to solve it! It appears to have been intended as a compliment to the chess problem (and tour) composer P. C. Taylor. It may help to know that three of the I's are the first person pronoun.

G I D E O R E O L K O A E Y E T E N L L N R U N I E C F M N A H I A O I I W S I H T H K O G H U T U E M H O T O H U E R A T G R

V D N E Y A I E D H O M A R W T I E N A M N D T W S E S P S E I C P S R S T L E L E N S S A T D R E T Y F M H A H A O E I Y N H

† 4: The final example, above right, is an improved version of an idea I first used on a 9×9 board (in Chessics #22, p.67, 1985). This contains its own clue; it is a closed path in which, to read the quotation correctly, one sequence of six letters has to be traversed twice (i.e. the same sequence of letters occurs twice).

« Cryptotours

The presentation of a knight's tour as a series of syllables or words written on the cells of the chessboard so that, when read in the order of the tour, they reveal a verse or message I call a cryptotour. This type of puzzle seems to have been a nineteenth century development, and its zenith was probably reached around 1875 when such tour puzzles were as popular as crossword puzzles became in the next century. If the tour is known the message can be read, and if the message is known the tour can be traced. Each encodes the other. Usually neither is known and it is a question of reconstructing both, using clues provided by each aspect to assist in the unravelling of the other.

† 5: The following is the earliest example I know of and appeared in the first chess magazine Le Palamède June 1842 (quoted in Cahiers de L'Echiquier Français 1930). It gives a 'squares and diamonds' tour, one word per square, revealed by a specially composed verse.

qu'au	d'allure	pourtant	et	coursier	fuis	fois	tous
mais	blanc	meme	et	quatre	en	ne	ta
toujours	tremble	noir	peut	trace	ton	sens	seize
monde	faire	change	point	galoper	pas	propre	repasse
parvenir	abime	cheval	franchir	marqué	tu	course	du
qu'un	d'un	toi	s'ouvrir	doivent	l'ourse	ainsi	verrais
un	veux	chaque	voila	tes	au	midi	sa
degré	ce	pieds	qui	jusqu'a	borner	sous	but

Further French examples, by Jules de Poilly, appear in Le Palamède in 1844 and 1846, using the tours of de Moivre and de Mairan. His poems are specially composed, to consist of eight lines each of eight words.

Il faut que ton cheval sente toujours l'entreinte, Prends soin de modérer sa trop bouillante ardeur, Car tu ne pourrais pas sortir du labyrinthe Si tu ne conservais le sang-froid protecteur; Le sort de Phaeton, ou le destin d'Icare Viendront t'atteindre, ami, si tu descends trop bas; En remontant trop haut par malheur on s'egare; Que le juste-milieu te sauve du trépas!

Galopez en tous sens, en avant, en arrière; Mais ne repassez pas par le meme chemin. Gagnez tantot le centre, et tantot la frontière, Si vous voulez mener l'affaire à bonne fin. Il faut, vous le savez, visiter chaque case, Sans en oublier une; et si votre Pégasse Se montre, en bondissant, rétif à votre voix, Dormez, vous le ferez courir une autre fois.

Another example, showing an irregular open tour, using a 64-syllable poem appears in A. Canel's Recherches sur les jeux d'espris (1867).

Examples in German were given by A. Herma Rösselsprunge aus deutschen Dichtern (1849). The Cleveland Library Ohio has provided me with photocopies of hand-written versions of ten cryptotours on the four-handed chess board, from Illustrierte Zeitung 1852 and 1863, which lacking German I have not been able to decipher. Also from Cleveland are copies of 41 'Rosselsprung' cuttings from an unknown source printed in 'Fraktur' type. The verses are from contemporary German poets, including E. M. Arndt (1769-1860), K. E. Ebert (1801-1882), G. F. Daumer (1800-1875), H. Heine (1797-1856) and F.Ruckert (1788-1866). Most of those that I have decoded show tours with maximum axial symmetry, having only four asymmetrically placed links (or three in open tours), indicating a systematic study of this topic. H. Schubert's Mathematische Mussestunden (1904) gives a three-dimensional (4×4×4) cryptotour.

† 6: The craze for cryptotours seems to have reached Britain only in the 1870s. The first English examples I have found are 16 given in Howard Staunton's chess column in the Illustrated London News 1870–1874. The first of these has one of the most amusing chess-related verses I have come across. The author is not mentioned; perhaps it was Staunton himself (he was a literary man; a one-time actor, and editor of an edition of Shakespeare).

sor	to	king	good	say	luck	loy	eth
and	moth	a	soon	dis	our	to	bad
place	ry	church	his	force	is	hat	al
er	queen	him	wight	he	to	may	truth
man	his	and	and	chess	es	knight	op's
a	sneer	the	and	un	lawn	of	tates
cas	that	at	less	pawn	no	bish	lant
eth	faith	tles	hath	the	gal	in	love

His subsequent examples employ a wide range of literary quotations, especially from Walter Scott and Shakespeare, and even include examples in French (de Musset), Italian (Dante) and Macaronic verse (Porson). None of the verses selected however are on subjects particularly relevant to tours. The popularity of these puzzles, as judged by the long lists of solvers, was great (this was before the invention of crossword puzzles) and it led to their appearance in other columns. There was a long series of 'Knightly Peripatetics' conducted by 'E.H.' in the Glasgow Weekly Herald beginning in 1872, some of the tours being the earliest known magic two-knight tours. An article by 'J.B.D.' in The Leisure Hour 1873 gives a cryptotour using a poem by Scott. The last item in the Poems and Problems of J. A. Miles (1882) bearing the title 'The Retrospect' encrypts a passage from Ossian (alias J. Macpherson).

† 7: Four good examples, two apparently sent to a knight's tour tourney organised by the British Chess Magazine and the others from Sussex Chess Magazine and Brighton Guardian are given in Chess Fruits (1884) by T. B. Rowlands and his wife Frideswide F. Rowlands. The first is given here for solution.

Now	irt	ght	our	heb	seo	pat	red
Kni	veils	bri	hun	Our	cke	ier	fs
mb	tray	day	ast	tyf	tot	cour	hwa
ness	he	Fro	ght	che	yne	ix	Wej
das	ima	dark	Our	ard	our	nda	ver
Till	but	yst	ge	the	O'er	our	Thr
lgr	lea	lif	ave	hwa	onw	cle	rkw
eis	lon	api	hat	ar	etr	oug	ney

The verse of the second, which may owe something to Jules de Poilly's French examples, is:

With nerve of steel and heart of fire
A gallant knight did once aspire
To roam the land of black and white
And prove to all his powerful might.
He started from the King's domain
And back again to it he came
Without missing a single square
Nor resting twice on any there.

The third quotes a verse from Shakespeare (Merchant of Venice Act 3, Scene 2, lines 131-8). The last is a seasonal greeting. Later examples appear in Puzzles Old and New by Professor Hoffmann (1893) and in A. C. Pearson's Twentieth Century Standard Puzzle Book (1907).

One of the oddest of these encryptions, and certainly the longest, is a booklet by H. Eschwege, published at Shanklin, Isle of Wight, in 1896, with the title The Knight's Tour, In a continuous and uninterrupted ride over 48 boards or 3072 Squares. Adapted from Byron's 'Mazeppa'. Byron's poem is presented on a series of chessboards, one word to a square, commencing at f8 and proceeding to e2, whence to leap to f8 on the next board, and so on. One tour (provided by H. E. Dudeney) is used throughout, but on the 48th board it is modified to end at h1.

† 8: An interesting use of a cryptotour was made by H. J. R. Murray in British Chess Magazine (1917):

he	fall	dom	ne	the	thinks	then	he
from	man	speaks	the	wis	may	think	ly
and	fore	him	wise	ver	hap	or	first
game	fol	thinks	of	fool	at	chess	our
be	words	tious	same	the	in	speaks	all
ly	less	out	dom	cau	is	roy	of
its	plays	and	rule	the	game	so	life
hour	frets	thought	a	wis	of	still	al

Murray wrote: “My most recent investigations have dealt with the tour on the whole board in general. I have made some advance, but not sufficient to make it desirable to give any account of the lines upon which I am working. Under similar circumstances mathematicians of the sixteenth and seventeenth century were wont to publish anagrams which concealed the key to their work, so that they could prove their claim to priority if a rival investigator forestalled them in publication. May I adopt their custom and give a tour in the puzzle form of older chess magazines, which illustrates three maxima in a theory of the tour which promises to reveal the total number of tours possible? No excuse is necessary in this magazine for borrowing for the purpose a poem from one of its most welcome contributors in its early days — the late Professor Tomlinson.”

D. E. Knuth reports that Fred Rixey's Jumblegrams, New York (1927), a booklet of puzzles published by the Wright Press, consists of 30 puzzles on a 6×6 board. Often there are blank squares, so the path length may be less than 36. Answers are not given in the book, but were available on writing to the publishers. The first problem uses a quotation from Horace.

† 9: To conclude these verse cryptotours, I show here an example of my own on the 6×6 board. It requires a reentrant tour by a 'bison' which moves in {1,3} and {2,3} leaps! To help I will say that most of the tour is comprised of camel {1,3} moves, with only one zebra {2,3} move.

some	by	if	bison	may	've
or	chance	you	camel	ing	in
and	camel	chess	swith	eye	or
son	son	sgo	white	seen	you
stripe	bison	to	jumps	swith	mak
lar	black	humps	son	zebra	set

« Crossword Puzzles

Another literary application of knight's tours is in the construction of special crossword puzzles. The most interesting example I have seen was by 'Alban' (Will Scotland) in Crossword, the magazine of The Crossword Club, in 1988. This presented the first verse of Jabberwocky, which happens (?!) to have 100 letters, in the form of a 10×10 knight's tour. The minor point that there are no actual words across and down was overcome by arranging that one letter be omitted from each answer before being entered on the diagram. This is a standard ploy among crossword buffs, known as 'Letters Latent'. Thus for example the fourth clue across was "Cask of wine the French knocked back around noon — boring!" The answer to this is TUN-N-EL, from which the Ns are omitted, giving TUEL to enter. Some letters remained unclued. The omitted letters, astonishngly enough, spell out the title of the book from which the quotation is taken.

In reply to a request in Crossword in 1992 I received several replies about crosswords with chess-move themes that appeared in the BBC Radio magazine The Listener. Derek Willan particularly recalled one that appeared "about 15 years ago and involved tracing a knight's move around the six faces of a cube". D. G. Mockford mentioned crossword 1020 by Pipeg (13 x/1949) in which clues were entered by following routes taken by knight, rook and bishop, and 1024 by Cocos with moves in knight paths. This had the title 'Knight's Move – IX' presumably the ninth in a series! H. J. Godwin listed 18 later examples. I reproduce here his list of numbers, dates and composers for any reader who may wish to follow up the references (1306 v/55 Gib, 1348 iii/56 Wray, 1378 x/56 Halezfax, 1386 xii/56 Sugden, 1468 vii/58 Halezfax, 1590 xi/60 Wray, 1632 ix/61 Wray, 1672 vi/62 Aeschylus, 1677 vii/62 Wray, 1709 ii/63 Doghouse, 1714 iv/63 Chabon, 1783 vii/64 Wray, 2150 viii/71 Sabre, 2407 i/77 Leiruza, 2428 vii/77 Leiruza, 2512 viii/79 Leiruza, 2900 v/87 Adam, 2980 xii/88 Casein).

Though the magazine has closed, 'The Listener Crossword' still appears in The Times on Saturdays. David Pritchard sent me a cutting of 3353 by Wolfram (13 April 1996) that uses a 12×12 knight's tour. Words are entered along the tour, commencing at the top left corner, one letter to each square. Further, by reflecting the tour in the principal diagonal a second series of words is spelt out along the tour! How some of these remarkable feats of word manipulation are accomplished I'm not too sure, but they certainly carry on and advance the tradition in an admirable way.

« Solutions

Polygraphy: Diagrams of the tours:

‡ 1: Staunton: 'The knight moves from one corner to the opposite on a rectangle of six squares.'
‡ 2: Cahiers: 'La beauté tend a l'unité tandis que la laideur est multiple: Barbey d'Aurevilly'.
‡ 3: McGuffey: 'When I take the humour of a thing once, I am like your tailor's needle — I go through'.
‡ 4: Omar: 'Where destiny with men for pieces plays, hither and thither moves and mates and slays' (from the Rubaiyat of Omar Khayyam, translated by Fitzgerald). The letters HITHER are traversed twice (shown by the darker lines in the diagram).

Cryptotours:

‡ 5: Le Palamède 1842:
Franchir chaque degré d'un monde blanc et noir
Galoper en tous sens du midi jusqu'a lourse
Voila ce qu'un cheval peut faire; mais pourtant
Quatre fois seize pas doivent borner sa course
Au but ainsi marqué toi qui veux parvenir
Tremble qu'au meme point ton coursier ne repasse
Tu verrais sous tes pieds un abime s'ouvrir
Change toujours d'allure et fuis ta propre trace.

‡ 6: Staunton:
The man that hath no love of chess
Is, truth to say, a sorry wight,
Disloyal to his King and Queen,
A faithless and ungallant Knight.
He hateth our good mother church,
And sneereth at the Bishop's lawn,
May ill luck force him soon to place
His Castles and estates in Pawn!

‡ 7: T. B. and F. F. Rowlands (the tour is a magic one, 12o by Jaenisch):
Our life is but a pilgrimage,
From birth unto the bier,
We journey onward as the Knight,
Our pathway never clear.
O'er checkered course of sixty-four,
Through ways that lead astray,
Now bright, then dark, we travel on,
Till darkness veils our day.

‡ 8: Murray (verse by Tomlinson)
The wise man thinks before he speaks
And words of wisdom from him fall.
The fool speaks first then haply thinks
Or he may never think at all.
So in our royal game of Chess
The rule of life is still the same.
Folly frets out its thoughtless hour
And wisdom plays a cautious game.

_

‡ 9: Jelliss
If in chess you may chance to set eyes on
Some camels with black and white stripes on
Or zebras with humps making camellar jumps
You've seen bison, or bisons, go by son!

_

_