GOOSE THIGHS REHASHED
by Christopher McManus
Word Ways, 1994
One day during his last illness, my father suddenly remembered a card trick from his boyhood. The trick involved the Latin mnemonic MUTUS NOMEN DEDIT COCIS. Everyone was astonished that a nonsense phrase had lain dormant in his mind's deepest recesses for more than sixty years, then sprung serendipitously to his weakened senses.
The same card trick was explained by Ross Eckler in the August 1968 issue of Word Ways. He provided an alternate English mnemonic, BIBLE ATLAS GOOSE THIGH. Both mnemonics allow one to guess the placement of a particular pair of cards in a tableau of five columns and four rows. Eckler further suggested the mnemonic LIVELY RHYTHM MUFFIN SUPPER SAVANT to find a pair of chosen cards in a 30-card tableau of six columns and five rows. In the February 1972 Word Ways, Eckler extended his results to 42 cards in a 7-by-6 tableau, using the Webster's Third sanctioned mnemonic MEACOCK RODDING GUFFAWS TWIZZLE RHYTHMS KNUBBLY.
In the present article, my own goal is twofold: (1) to search the realm of seven-letter candidates systematically in order to find how rare 7-by-6 tableaus are, and thereby to find the easiest mnemonic to teach younger friends, and (2) to extend the card trick to triplets and quadruplets of each letter. The latter approach has apparently never been considered before.
To appreciate the word patterns demanded by the mnemonics, it is important to know the card trick itself. Knowledgeable readers can skip the rest of this paragraph, wherein I reprise the basic steps. Select any 20 cards from an ordinary deck for the four-row trick (alternatively, select 30 cards for the five-row trick, or 42 for the six-row one). Place the selected cards face down in separate two-card piles. Ask a volunteer to pick any two-card pile, to look at and memorize those two cards, and then to replace them face down. The volunteer should then form one stack from the piles, in any way that does not break up any two-card pile. The conjurer then playes the cards face up in four, five or six rows, according to the mnemonic. Each successive pair of cards is placed in positions corresponding to a single letter in the mnemonic. Suppose the mnemonic is RODEO DUMMY YELLS TRUST. You could put the first two cards in columns two and five on row one (corresponding to the O's), or in the rows and columns corresponding to the two R's, or any of the other eight repeated letters. Put the second pair, and each succeeding pair of cards, in positions corresponding to some repeated letter. Finally, ask the volunteer to indicate which row or rows contain his or her selected cards. Whichever answer is given will determine a unique repeated letter, and the selected pair of cards will lie at the positions of that doubled letter.
I first systematically studied the problem of fitting 21 pairs of letters into a 7-by-6 tableau. As Eckler suggested in his article, vowels are one key to an efficient search for six-row solutions. If we restrict the vowels to AEIOU, then there is no way each of six words can have at least two vowels apiece, and still have no vowel appear more than twice. At least one word must have no AEIOU vowel, or at least two words of the six must have exactly one AEIOU vowel. To further expedite my search, I subsorted the candidate words by number of vowels and repeated letter. Thus, PSYCHIC fell into list 1-C. Only one word from any subcategory can fall into any solution. One can also use the symmetry of solutions to search only for words which alphabetically follow previously-found positions.
My corpus of words includes some 237,000 words of lengths between five and fifteen letters. 302 solutions emerged, several of which were particularly interesting. Three consisted entirely of words found in both the Official Scrabble Players Dictionary and the Random House Concise Dictionary:
bombard, gruffly, hunched, jotting, pimples, skyjack
bowknot, crackup, distaff, hyphens, mumbled, wriggly
checkup, gruffly, symptom, tankard, wishing, wobbled
Only three solutions used the letter X:
bedbugs, cachexy, drizzly, fawning, flummox, trowths
bopping, chymics, gruffly, handled, waxworm, subtext
excysts, knubbly, paddler, pickoff, through, waxwing
Another seven used the letter V:
badging, chaffed, nymphly, ptotics, vulvars, webworm
bombard, chugged, flivver, nymphly, sunfast, topkick
budging, chuffed, nymphly, ptotics, vulvars, webworm
budging, convect, flyblew, foppish, thrummy, vawards
buzzard, clutchy, gyppers, longbow, vivants, whiffed
crunchy, gloving, puzzled, symptom, vawards, whiffet
cutback, guzzled, offhand, phytyls, proverb, skiving
EXCYSTS, KNUBBLY and PHYTYLS occur in Webster's Third. NYMPHLY is a Webster's Second word. VIVANT occurs in Hamlyn's Complete Crossword Dictionary.
Eckler suggested two shorter mnemonics with multiple food images, the GOOSE THIGH and MUFFIN SUPPER phrases. The most alimentary six-row mnemonic is THIRSTY PUMPKIN BLACKLY SCUFFED HANGDOG WEBWORM.
Longer mnemonics are most memorable if they form a phrase or sentence. Images formed by the 42-letter solutions include:
Bopping chymics gruffly handled waxworm subtext
Thrummy knocker whapped Fascist, budging blowfly
Wobbler, priding symptom, hatched knuckly guffaws
Jumpoff dogsled, rubbing peccant rhythms jazzily
Fifthly, nudnick whammed scraggy topwork puzzles
Each of thes above tricks involves two-card piles, or equivalently pairs of letters. How about stepping up to three-card or four-card piles? We can first solve the trick where letters are repeated triply. The major complication will be that any given letter may have its three occurrences all in the same row, in two rows, or in three different rows. The simplest possible arrangement to satisfy these conditions is 21 cards in a 7-by-3 tableau. I was surprised to find only 19 solutions of this tableau in a large word list. These include three entirely formed by Official Scrabble Players Dictionary words:
giddied, nooning, gessoes
ninepin, pallial, asepses
pontoon, apparat, enterer
with REENTER, TERREEN or TERRENE possible substitutes for the last word. Another solution is my favorite: SINNING, GOOGOLS, ALALIAS. To use this mnemonic you might lay down two successive cards in the places of the I's in SINNING in the first row, followed by a cardin the third row in the positon of the I in ALALIAS. You would lay down successive triplets of cards in positions corresponding to repetitions of a single letter. Finally, you would ask the volunteer to point to the row or rows in which his cards appear. Any of the seven possible answers determines a unique trio of cards.
A variation of the triplet problem collects 14 piles of three cards each, and arranges them into four rows. Half the rows have eleven cards each; the other half have ten cards each. To solve this problem I needed assistance from two large word lists provided by Ted Clarke and Len Gordon. The 62 solutions included the three below. The first two include only uncapitalized noncompound words found in Webster's Second or Stedman's Medical Dictionary; the first and third present distinctive images.
unbragging, pineapples, buccoclusal, bitterroots
worrywarts, wagglingly, saddlenosed, cytocinetic
unguttural, pig boiling, stopperless, bed and board
To confirm these solutions for yourself, choose one and pick any combination of rows up to three. For example, take rows two and four in the first solution. Three E's are divided between rows two and four. The chosen cards would correspond to these three positions.
Next, we can tackle the extension involving piles of four cards. The simplest tableau to encompass all combinations of one, two, three or four rows is a 15-by-4 arrangement. Since this entails 60 cards, exceeding the number in a standard deck, any solution is probably only of theoretical interest. Two working solutions are given below.
No foursome of simple fifteen-letter words could be fitted together to form a tableau. Instead, the following mnemonics replace one or two of the four fifteen-letter words with phrases, but otherwise meet the challenge of an easily-remembered mnemonic for the four-card pile challenge:
nonproportional, flibbertigibbet, daunt added fluff, plugs grasp guess
dactylalgically, nonregenerating, spectroscopists, you hunt punch cup
For either mnemonic, ignore all spaces. Now pick any row, any two rows, any three rows, or all four rows. There is a unique letter found only in that combination of rows! Each row possesses a special property: it contains exactly one letter repeated four times, at least one unrepeated letter, and exactly eight different letters altogether. Of a total 11,500 fifteen-letter words in the word lists of Gordon, Clarke and myself, only 340 words met these three conditions.
The above mnemonics exploit all possible locations of two (or more) cards in n rows, and as a consequence the fewest possible number of words are used in the mnemonic. Eckler pinted out that this is not a necessary part of the card trick--one can decide not to use certain pairings. For example, using the mnemonic WAD SET CON DEN CAT SOW, no card-pairs appear in a single row, and there are no card-pairs in the first and second, first and third, second and third, fourth and fifth, fourth and sixth or fifth and six rows. Why not try for the ultimate goal, a mnemonic which enables one to predict any one of 26 pairs from a full deck of cards? (It is a pleasant coincidence that such a mnemonic would use each letter of the alphabet exactly twice.) If thirteen four-letter words are used in this mnemonic, one word must be vowelless, with the vowels AEIOUY rationed one apiece to the remaining twelve. Furthermore, the two Q's must be used in words each using one vowel. I found a solution using the Official Scrabble Players Dictionary: QOPH, JAZZ, HYMN, JINX, VEXT, WORD, BUFF, PLED, BYRL, KICK, CWMS, QATS, VUGG.
Ted Clarke created a two-dimensional version of the mnemonic. If one lays out ten card-pairs according to the mnemonic at the left below, the proper pair can be identified by specifying in which unique row or column the two cards are located (the L-pair is not confined to a unique row or column, but since this is the only such pair in the array it can be identified too). The second and third arrays (which deploy nine pairs and eight pairs, respectively) work perfectly.
L O O N Y R O M P E R C O C K S T U N T L O G G E D P O M P S C U F F L I M P I D L A M A I C I L Y L E E KLike other successes, the BIBLE ATLAS GOOSE THIGH algorithm begs for variants and extensions. The alternative conditions and solutions that I have explored in this article all flow logically from the original problem. They also form appealing card tricks on their own. Doutbless these extensions will suggest yet further extensions to the reader.