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For the purpose of an arithmetical puzzle I am devising, I require a nine-letter legitimate (i.e. acceptable to Scrabble players) English word such that at least one permutation of all the letters of each of the, as many as possible, 511 combinations (without repetitions) of one or more of its letters is itself a legitimate English word.

I have been told of a word with just over 100 such combinations. How much better can I get?

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    $\begingroup$ So the dictionary you are using is Merriam-Webster, is that correct? $\endgroup$ – hexomino Jun 1 at 18:11
  • $\begingroup$ @hexomino: That would certainly do. $\endgroup$ – Bernardo Recamán Santos Jun 1 at 18:17
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    $\begingroup$ I don't think permutations is the right term here. The number 511 suggests that letters are either in or out, and the order of the letters is preserved. $\endgroup$ – Daniel Mathias Jun 1 at 18:35
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    $\begingroup$ I'm sorry, I don't think I understand. Are you asking for a 9 letter word where any of its 9 letters can be changed to a different letter and make a new word? Or are you asking for a word that can be anagrammed/scrambled/letter banked into over 100 smaller words? Or something else, perhaps? $\endgroup$ – Reibello Jun 1 at 20:39
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    $\begingroup$ I think I understand now. If the letters include {A,E,R,T}, we should count AERT (rate/tare/tear) AER (are/ear/era) AET (ate/eat/eta/tae/tea) ART (art/rat/tar) ERT (ret) AE (ae) AR (ar) AT (at/ta) ER (er/re) ET (et) and A (a), a total of eleven combinations that provide at least one word. $\endgroup$ – Daniel Mathias Jun 1 at 21:10
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The question did not state no computers, so I guess this answer is as good as any other.

The word posterial

...gives...

Aeolis Aeolist Al A Aletris Alois Ao Aotes Apios Apis April Ariel Aries Aristol Aro Art Artie As Ati Elaps Eli Elia Elias Elsa Eros Ila Io Ira Isleta Isoptera Israel Itea Ito Lao Lar Lari Lars Lasi Latris Lear Leipoa Leo Lepiota Leporis Ler Leto Lise Lo Lois Lora Lot Lota Olea Oreas Orias Os Ostrea Otis Pali Paris Pat Pelias Peoria Perla Piast Pilate Pilea Piro Poa Poales Polaris Pole Poria Porites Portia Protea Pteris Rais Restio Rio Rita Ro Roist Rosa Rosalie Rotal Rotse Saite Sart Sepiola Sept Septi Seri Sia Sir Sita Sol Solea Solera Spar Taos Tape Trias Trio Triops ae aer aero aes ai aire ale alepot alert aliptes alisp alist alit alite alop alp alpist also alt altrose aorist ape aper apert apio apiose aport apostil apostle apse ar areito ariose ariot arles arse arsoite aslop asop asp asper aspire asport ast aster astor at ate atelo ates atip atle atop atrip el elator elisor elops els elt epistlar epos er ers es esparto estop estriol iao ie ilot iota ire is islet islot iso ist it iter lapser las last laster leap leapt least lerot lerp lest li lier lip lirate lis lisp lisper list lister lit lite liter litra loir loiter lop loper lore loris lors lose loser lost lotase oat oe oer oes oil oiler ope opera opiate opt orate ort osier ostial pa pair palster palt palter par parol parole paroli parolist parotis part partile past paste pastel paster patesi patio patois patrol peai peloria pelt per peri peril perit pert pes pet petiolar peto petrol petrosal pi pia piaster pie piet pile piles pilose pilot pir pirate pirl pirol pise piso pist pistle pistol pistole pit plait ploat plot plote po poi poil poise poiser poitrel polis polite pore porite port portail poser posit post postal poster poter prest prisal pro proa prolate psi psora pst rapist ras rate relost reposal reps resoap resplit rest ret risp rist rit rosal rota rote sailor salp sao sapote satire seit sepia set sie siper sitao slart slept slirt slop sloper slorp slote sop sopite sori sort sortal sot soterial sotie spalt spart spilt spirale spiro spirt spolia spoliate spor sporal sport sproil st steri stola stopa stroil tailor te tie tiple tirl to toe topi topsl trip tripal tripel posterial

which gives a score of 389. This is using the Merriam-Webster dictionary. No word is an anagram of another, thanks to @hexomino for pointing this out.

Excluding names, the same word is still optimal:

a ae aer aero aes ai aiel ail air aire airt aisle ait al ale alepot alert aliptes aliso alisp alist alit alite alo aloe alop alose alp alpist also alt alto altrose aorist ape aper apert apio apiole apiose aplite aport apostil apostle apse apt ar areito ariel aril ariose ariot arise arist arite arles arose arse arsoite art as aslop aslope asop asor asp asper aspire asport ast aster astor at ate atelo ates atip atis atle atop atrip earl el elator elisor elops els elt epistlar epos er eros ers es esparto espial estop estriol iao ie ilot io iota ire is isle islet islot iso isolate ist it iter laiose lapse lapser lar las last laster latro leap leapt least lei lepra lerot lerp lest li lier lip lipa lirate lis lisp lisper list lister lit lite liter litra lo loir loiter lop lope loper lora lore loris lors lose loser lost lot lotase lote oar oast oat oe oer oes oil oiler ope opera opiate opt or oralist orate ort os osier ostial pa pair pais palster palt palter par parol parole paroli parolist parotis part partile past paste pastel paster pate patesi patio patois patrol peai peloria pelt per peri peril periost perit pert pes pest pet petiolar peto petrol petrosal pi pia piaster pie piet pilar pile piles pilose pilot pir pirate pirl pirol pise piso pist pistle pistol pistole pit plait ploat plot plote po poi poil poise poiser poitrel polis polite pore porite port portail portia poser posit post postal poster poter prest priest prisal pro proa prolate protea psi psora pst rais rapist ras rasp rate realist reis rel relais relost reposal reps resoap resplit rest ret rio riot ripost risp rist rit rosal roseal roset rota rote sai sail sailor sale salite salp sao sapote sarip sart satire seit sepia set sie silo siper sir sitao slart slept slirt slop sloper slorp slote sol sop sopite sori sorite sort sortal sot soterial sotie spalt spart spilt spirale spiro spirt spite spoiler spolia spoliate spor sporal sport sproil st steri stola stopa strial stroil tailor taise tapis te tie tiple tirl to toe toi topi topsl trip tripal tripel

gives a score of 369.

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  • $\begingroup$ @CarlLöndahl I think the counting isn't right because the max suggested is 511. I think the question is asking "how many subsets of the letters can be rearranged to form a word?" $\endgroup$ – hexomino Jun 1 at 21:18
  • $\begingroup$ This is exactly what I am counting. The wordlist is a bit strange since it includes single letters. I will try with Merriam-Webster instead. $\endgroup$ – Carl Löndahl Jun 1 at 21:20
  • $\begingroup$ @hexomino, Right, I counted all permutations for each subset. $\endgroup$ – Carl Löndahl Jun 1 at 21:45
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    $\begingroup$ Nice, that seems like a difficult score to beat $\endgroup$ – hexomino Jun 1 at 22:32

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