When Douglas Hofstadter took over Martin Gardner\u27s column in Scientific American, he changed the name from MATHEMATICAL GAMES to METAMAGICAL THEMAS. The new name is an anagram of the old one, that is, a permutation or reordering of the same set of letters. This naturally suggests the question of how easy it is to find an anagram on these letters which  makes sense

Golomb, Solomon W.

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20 AMALGAMATE, CHEMIST! SOLOMON W. GOLOMB Los Angeles, California When Douglas Hofstadter took over Martin Gardner's colum in Scient ific America.n, he changed the name from MATHEMAT ICAL GAMES to METAMAGICAL THEMAS. The new name is an anagram of the old one, that is, a permutation or reordering of the same set of let­ters. This naturally suggests the question of how easy IS it to find an anagram on these letters which "makes sense". Some sets of letters (such as XJRMQ) will form no English words or phrases at all, though the die-hard may find the call letters of a Mexican radio station' Some sets of letters (such as those in the word FLANK) will form only one word; some will form exact­ly two (N IGHT, TH ING); some will form several (POST, POTS, SPOT, STOP, TOPS, OPTS). How many "meaningful" expressions can in fact be formed from the 17-letter set AAAACEEGHI LMMMSTT, the letters in MATHEMATICAL GAMES? First, the purely ma thema tica I question: how many distinct per­mutations of these 17 letters are there, whether they make any sense or not? If all 17 letters were different (which they clearly are not), the number of permutations would be 17 1, pronounced "seven teen factorial," and defined to be 17 1 17x16x15x14x13x12x 11xI0x9x8x7x6x5x4x3x2xl 355,687,428,096,000. Because there are repeats among the letters, we must divide this number by the num­ber of ways of permuting identica 1 letters among themselves (since interchanging identical letters produces no visible change in the resulting sequence). The four A's can be permuted in 4 1 = 4x3x2xl or 24 ways, the three M' s in 3x2xl or 6 ways, the two E' s in 2 ways, and the two T' s in 2 ways. Thus the total number of dis­tinct sequences of the 17 letters in MATHEMATICAL GAMES becomes 171/(4!x3!x2~x2~) 617,512,896,000. This number is increased if we have the opt ion of introd uc ing spaces between words and punc­tuation marks wherever we wish in the sequence. How many of these mathematically distinct sequences are likely to make any sense as English? This is a question for Information Theory, which reformulates it as follows: What is the entropy of written English, regarded as a source of letters of the alphabet? (Entropy is a measure of the permitted degree of randomness.) If written English had an entropy as large as log 226 = 4.7.. bits per letter, that would mean that essentially every sequence of let­ters is "meaningful". We know this is far from the case, and the best estimates of the entropy of written English are in the range of one to two bits per letter. This would suggest that there are 2 2Nbetween 2 Nand meaningful English messages which are N let­217 234ters long, and in particular, between = 131,072 and = 17,179,869 for sequer the upper for large mutations es, for la with the t The let€ M 's but r 0' s, etc. ful" seque be inserte be expres~ is sensitiv constitutes "words" w] am awa we are to PHYSICAL; and schem, come them, words, va" proper nan and more I To illus anagrams form for t of serve " ial mope p Reve~ My servi sections 01 ogy, soror mathematic LEMMAS; tl MAGMA, MA with CAME: THETA, Sl< vicious ad letics sect: ATHLETlCS HEAT, HAM and MEASL a ti ve anag For seve employed 1 use of per scrambled the sender 's colum in nCAL GAMES 1 of the old set of let­lSy is it to 19iish words call letters ch as those form exact­POTS, SPOT, can in fact the letters jistinct per­f make any they clearly pronounced 15x14x13x12x there are by the num­,elves (since ange in the 41 = 4x3x2x1 wo E' s in 2 nber of dis­MES becomes increased if sand punc­are likely Informa tion entropy of 1e alphabet? domness.) If :: 4.7 .. bits lence of let­lse, and the n the range l. t there are 1 are N let­234and 21 17,179,869,184 English "messages" 17 letters in length. I suspect, for sequences of 17 letters, the truth is closer to the lower than the upper estimate. In any case, this would only be approximated for large values of N, and would not correspond to only the per­mutations of a single set of letters. However, most of these messag­es, for large values of N, would be permutations of sets of letters with the typical letter frequencies of English. The leters in MATHEMATICAL GAMES are not very typical: three M's but no N's, R's, or D's; four A's but only two E's and no O's, etc. My own estimate is that there are perhaps 3000 "meaning­ful" sequences of these letters (where punctuation and spaces can be inserted at will, and only fragments of ideas or phrases need be expressed), give or take a factor of 5. And even this estimate is sensitive to how strict or liberal we are with our notion of what constitutes "English". Thus, METAMAGICAL THEMAS consists of two "words" which are not in any standard English dictionary of which I am aware. METAMAGICAL is a Hofstadter coinage whose meaning we are to deduce by analogy with METAMATHEMATICAL and META­PHYSICAL; and THEMAS is an improper plural of THEMA. (Theme and scheme lead to themes and schemes; but from thema and schema come themata and schemata.) As we allow more and more foreign words, variant spellings, abbreviations, initials, contractions and proper names, the entropy of "English" increases, and we get more and more "meaningful" anagrams. To illustrate what is possible, I have listed 100 "meaningful" anagrams of MATHEMAT ICAL GAMES. I have created a new literary form for this purpose, which I will call serve verse. An example of serve verse can be called a mope poem. A brief, self-referent­ial mope poem is: Reves: Sever verse, / Veers; serve. My serve verse creation on MATHEMATICAL GAMES is divided into sections on the general themes of metamathematics, geology, zool­ogy, sororities J athletics, culinary arts, and medicine. The meta­mathematics section is concerned with such things as THEMATIC LEMMAS; the geology section (or is it mineralogy?) with AMALGAMS, MAGMA, MALACHITE and STALAGM ITE. The zoology section is obsessed with CAMELS and MAMMALS. The sorority section has GAMMA, ETA, THETA, SIGMA and CHI at its disposal, and seems to contain a vicious ad feminem attack against one EMMA GAL E SM 1TH. The ath­letics section refers to TEAMMATES, GAMES, SETS, MATCHES and to ATHLETICS itself. The culinary section mentions STALE MEAT, STEAM HEAT, HAMS and CLAMS, and the medical section alludes to TEETH and MEASLEs. Readers are encouraged to invent even more imagin­ative anagrams of their own. For several centuries, one of the forms of cryptography actually employed to conceal the content of diplomatic messages was the use of permutation ciphers, where the letters of the message were scrambled according to a procedure known only (it was hoped) to the sender and the intended receiver. Without the correct rule, 22 as we have seen, the same letters might be unscrambled to reveal any "secret" message from MAGMA HEATS CL I MATE to SAM, THE MAG­lC TAMALE. Ma thema tical Games Metamagical Themas Thematic lemma saga 1 get the lemma - a scam Meet math as magica 1 Elastic math game, Mal Magic sheet: lama ma t The same magIcal mat Image chats at lemma G. 1. lemma ta schemata Ama Igama tic Themes The Magmatical Seam Ama Igama te, chemist ~ Misama Igama te, Chet I Chemist ate amalgam Alchemist a te magma Magma hea ts climate Mahatma I s cl imate, e. g. Magi select mahatma Michael, taste magma Ma lachite set magma Ma laes thetic magma Sta lagmite came, ham Ah, magmatic Ma Itese Almagest: teach imam Mega lith caste, Mama This Camel Ate Magma Magma? Camels Hate It Get this, Ma 'am: a camel Get this, ace: a mammal' Tigh t as a camel, Emma Itaches? Get a mamma 1 ~ Camel meat? l'm aghast' Aghas t, Mac? Mea ltime ~ As a camel might mate 1 get a chaste mammal I get mahatma's camel Che agitates mammal Castiga te mammal, eh? Aceta te mammal? Sigh! Cage a theist mamma 1 Ha ttie cages mammal Tia cages the mammal I teach stage mammal That I s ice-age mammal Mamma 1 at "eigh t aces" The camel's gait, Ma'am The ice mammal's gate Age maims that camel Ice ages that mammal Camel stigmata? Ahem ~ LegIt schemata, Ma' am Hag Claims Teamma te "Ha te Mate," Claims Mag Chi Gamma's late mate Theta Gamma's malice Asthmatic Emma Gale (Emma Gale Smith: a cat?) Met me, claims Agatha Me? Clammiest Aga tha! Gamma Chi stalemate Eta Chi Gamma metals Mag's Mealtime Chat Mag's Calm Teatime? Hal Mastica te ham leg, Ma Magic ham? Stale meat l Chi Gamma: taste meal ~ Teatime: ham, clam, gas Stage claim: ham meat Malt, sage ethic, Ma' am Came as tamale might Sam, the mag ic tama le Mama Mia' The Last Egg' Ethical, Sam? Get Mama I Aim, match t tag measle I calm the stage-mama Mica teeth amalgams Asthma:' age claim met Ath letic Michaelm A late g< Cla im ga Aim, A1: Sham cag Met Lhas Mime act. 1 match « Team galT Hi t game EdItor's sentences mgs in a very of an an phrases ( Put in 01 the origii graphers lec tam sounds me OMN This last IS USUo lems bets: IS a lly easy ddh Thou abou Mart. les. to tl is d~ Igma· er, qUIZ buse~ IS a 1 hope d to reveal I, THE MAG-a scam Mal 1at ata m ~ t ~ a e.g. rna ~se rna :e It lammal ' immal' time! mmal tal igh t al mal aces" gate Imal a 'am ns Mag .lice . a cat?) l.tha I ~ta ls ne? Ha ~ meat l lm, gas Ma'am imale Mama l ·mama n met 23 Ath letic Games, Ma' am Thalamic Team Games Michae lmas team tag 1 gash calm teammate A late game mismatch A lame gate mismatch Claim gate at Thames Time Malaga matches Aim, AL game, set, match Lit matches age Mama Sham cage title, Ma'am Cla im tag? Shame, team! Met Lhasa magic team Act the lama ism game Mime act: Hamlet saga Ham act; get me sa lam i 1 match eaglets. Ma' am I tag the camel's mama Team game; Cal/Smith 1 get Tech's alma mama Hit game castle, Ma' am Agi le mama cast them Editor's Note; One can ask what is the likelihood that any of the sentences given above have somewhere appeared in English writ­ings in a meaningful (not wordplay) context. Obviously, this is a very stringent (and unverifiable I ) measure of the plausibility of an anagrammed phrase; it is quite possible that none of these phrases (save the already-noted METAMAG lCAL THEMAS ~ would pass. Put in other terms, it is not difficult for the reader to pick out the original phrase on which the anagrams were based. Crypto­graphers using permutation ciphers would be well-advised to se­lect a message using convoluted grammar, with an anagram that sounds more plausible than the intended message. OMNI GAMES This paperback anthology of the best brain-teasers from the last five years of Scot Morris's Omni magazine Games column is devoted to an ama zing variety of subjects.' besides the usual mathematics and logic puzzles, it contains physics prob­lems, geographical oddities, electronic calculator tricks, bar bets, juggling, and a tribute to Martin Gardner. The emphasis is on (1) problems that look difficult but can be solved eas­ily with the proper insight (aha l), (2) problems tha t look easy but contain hidden traps, and (J) problems that look difficult and in fact are, but which lead to curIOUS facts . Though many problems are golden oldies, Scot Morris writes about them in a lively and entertaining way reminiscent of Martin Gardner, and often succeeds in bringing in new ang­les. The book is enriched by reader comment and emendations to the original Omni material. A substantial part of the book is devoted to problems related to logology: four unsolved En­igma-style conundrums by Bishop Wilberforce, the Beale ciph­er, two tough spelling qUIzzes, the "world I S hardest" word qUIZ (containing much Word Ways material), mnemonICS, re­buses, anagrams, and the first crossv.1ord puzzle. The book is available from Holt, Rinehart and Winston for $11.95. Let's hope that this is the start of a long series. 

Amalgamate, Chemist!

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Amalgamate, Chemist!

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