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I know a set of six 4-letter words without repeating a letter (i.e. using 24 different letters).

Here is an example:

gasp, verb, jinx, flow, duck, myth

Are there sets of words with no repeated letters and having

  1. four 6-letter words?
  2. five 5-letter words?
  3. three 8-letter words?

Bonus: Find a set of words (non-repeating letters) of sizes 1,2,3,4,5,6.

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6 Answers 6

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Using words from the SOWPODS word list, it's possible to solve Q1 and Q2 but sadly not Q3. I checked this by creating a graph of k-letter isograms, connecting words with no shared letters, and searching for cliques.

Five 5-letter words

Lots of answers, all using obscure words: e.g.

brick, jumpy, vozhd, glent, waqfs
vibex, fjord, nymph, waltz, gucks
bling, jumpy, vozhd, treck, waqfs

Four 6-letter words

Lots of answers, some using slightly less obscure words: e.g.

jawbox, kvetch, flumps, drying
jumbly, dwarfs, kvetch, poxing
jawbox, fledgy, skrump, chintz

Three 7-letter words

Lots and lots of answers: e.g.

stumped, flyback, whoring
mucking, batfowl, zephyrs
jordans, phlegmy, fuckwit
overply, dumbing, thwacks

Two 10-letter words

blacksmith, gunpowdery
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Here's a set for the bonus question:

a
be
why
jump
flong (a type of mould used in relief printing)
tricks

The missing letters are:

dqvxz

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At the end of "Pangram Variations" (Word Ways, Feb. 1977, p.44), Ross Eckler gives the following examples:

For 1. (four 6-letter words): MUZJIK PEGBOX DWARFS LYNCHT

For 2. (five 5-letter words): FUDGY JAMBS PHLOX WRECK QVINT

Eckler says that Howard Bergerson came up with that set of 5-letter words in Webster's Second. His three-word example is of 7-letter words (JACKBOX FRESHLY DUMPING), suggesting that, even with a dictionary as large as Webster's Second, there is no set of three 8-letter words to be found.

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I have 2 possible answers for the four 6-letter words. Both were obtained by parsing a scrabble 6-letter word list, picking out words with few vowels, then checking to make sure there were 24 distinct letters.

blowzy, frumps, jading, kvetch
frowzy, jading, kvetch, plumbs

I attempted the same thing with five 5-letter words, but did not get a match.

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For the 1-letter bonus, there are two possible answers:

The first is that there are three different words:
A
I
O
Strictly speaking, these are the only one-letter words in English ("O" is an alternate spelling of the interjection "Oh").

The second answer relies on a more flexible meaning of "word": every letter can be regarded as a self-referential word. That is, as a word that identifies the letter, as in "C is for Cookie" or "The L Word". In this case, one can list 26 distinct words with no letters in common between any of them!

I'm currently working on the eight-letter question (and I want to do it without writing a search script), but as a hint to other solvers: there are only five vowels not including Y (which I just used as a word), so one of the eight-letter words will need to have only one "regular" vowel, and will probably need to use Y as a vowel. Some possibilities are BRIGHTLY GRYPHONS HYDRANTS PSYCHING SHREWDLY SPLOTCHY. It might be especially useful to use SCHMALTZ or SCHMALZY/SHMALTZY to use a Z.

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  • $\begingroup$ If I'm right that the only English 8-letter word containing a single vowel letter is strength, which isn't allowed because it's got two t's, then each of the three 8-letter words must have exactly two vowel letters. $\endgroup$
    – h34
    Jan 26, 2015 at 21:02
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    $\begingroup$ @h34 I think strengths is the only English 9-letter word with only one vowel, but there are at least a few other 8-letter ones. For example, schmaltz, which is in my answer. That's the only one I know of with no repeating letters, though. $\endgroup$
    – KSmarts
    Jan 26, 2015 at 21:27
  • $\begingroup$ Thanks for this. I stand corrected. The longest one-vowellers in SOWPODS are two with nine letters (strengths, tsktsking) and 20 with eight (borschts, pschents, schmaltz, shrights, sprights, strength, tsktsked, twelfths, and 12 of the form sch----s borrowed from Yiddish or German), so going by their list schmaltz is the only one-voweller with eight or more letters and no repeats. $\endgroup$
    – h34
    Jan 27, 2015 at 9:14
  • $\begingroup$ By the proper definition of a vowel as a phoneme, tsktsked has none at all! $\endgroup$
    – h34
    Jan 27, 2015 at 9:23
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Out of curiosity I looked at this:

First, get a word list, simplified to only have letters a-z

D=/usr/share/dict/british-english-insane
cat $D | cut -d\' -f 1 | \
    sed "s|Å|a|g;s|Ö|o|g;s|Ü|u|g;s|ö|o|g;s|é|e|g;s|á|a|g;s|à|a|g;s|â|a|g;s|å|a|g;s|ä|a|g;s|ç|c|g;s|è|e|g;s|ê|e|g;s|ë|e|g;s|í|i|g;s|î|i|g;s|ï|i|g;s|ñ|n|g;s|ó|o|g;s|ô|o|g;s|ø|o|g;s|ù|u|g;s|û|u|g;s|ü|u|g" | \
    tr '[A-Z]' '[a-z]' | sort -u > only_lower_a_z.txt

Next, remove from that list repeating letters, leaving heterograms:

grep -vE \ 
    "a.*a|b.*b|c.*c|d.*d|e.*e|f.*f|g.*g|h.*h|i.*i|j.*j|k.*k|l.*l|m.*m|n.*n|o.*o|p.*p|q.*q|r.*r|s.*s|t.*t|u.*u|v.*v|w.*w|x.*x|y.*y|z.*z" \
    only_lower_a_z.txt > just_heterograms.txt

The resulting file have these many lines ( = words)

wc -l only_lower_a_z.txt just_heterograms.txt
 486070 only_lower_a_z.txt
  87543 just_heterograms.txt

Easy now to see the counts and distribution:

cat just_heterograms.txt | tr '[a-z]' 'x' | sort | uniq -c | grep -n ^
 1:     26 x
 2:    330 xx
 3:   2114 xxx
 4:   7321 xxxx
 5:  14697 xxxxx
 6:  19269 xxxxxx
 7:  18225 xxxxxxx
 8:  13241 xxxxxxxx
 9:   7471 xxxxxxxxx
10:   3360 xxxxxxxxxx
11:   1122 xxxxxxxxxxx
12:    302 xxxxxxxxxxxx
13:     52 xxxxxxxxxxxxx
14:     11 xxxxxxxxxxxxxx
15:      2 xxxxxxxxxxxxxxx

Here's the first 5 from each see:

seq 15 -1 1 | while read LEN ; do \
    STR=`echo '................' | cut -c -$LEN` ; \
    echo -n "$LEN : " ; \
    grep "^${STR}$" _tmp2 | head -n 5 | tr '\n' ' ' ; \
    echo ; \
done
  15 : dermatoglyphics uncopyrightable 
  14 : ambidextrously benzhydroxamic dermatoglyphic hydromagnetics hydropneumatic 
  13 : amphigenously brachydontism bridgehampton chimneyboards chromeplating 
  12 : absorptively adjunctively adrenolytics adsorptively ambidextrous 
  11 : abolishment abridgments abruptiones absolutized achondrites 
  10 : abductions abductores abjections abjunctive abortively 
   9 : abducting abduction abductors abjecting abjection 
   8 : abditory abdomens abducens abducent abducing 
   7 : abderus abdomen abduces abducts abelson 
   6 : abdest abdiel abduce abduct abeigh 
   5 : abdel abdom abdon abdul abend 
   4 : abcs abdu abed abel abet 
   3 : abc abd abe abm abn 
   2 : ab ac ad ag ah 
   1 : a b c d e

See also : https://en.wikipedia.org/wiki/Heterogram_(literature)

EDIT: quick followup;

Strings of vowels

seq 5 -1 2 | while read V ; do  \
        PTRN=`for z in $(seq 1 $V) ; do echo -n '[aeiou]' ; done` ;  \
        echo -n "$V : " ;  \
        grep -c "$PTRN" just_heterograms.txt ;  \
        grep "$PTRN" just_heterograms.txt |  \
            head -n 10 | tr '\n' ' ' ;  \
        echo ;  \
    done
    5 : 1
    miaoued 
    4 : 26
    beauing beauish beauism codiaeum couaism coueism coueist cypraeoid douai euoi 
    3 : 832
    acquiet actious adeuism adieu adieus adieux aeolic aeolics aeolid aeolight 
    2 : 22499
    abdiel abduction abductions abeigh abeu abie abied abiegh abient abies

Without vowels

grep -v "[aeiou]" just_heterograms.txt | \
    tr '[a-z]' '.' | sort | uniq -c | grep -n ^
    1:     21 .
    2:    154 ..
    3:    287 ...
    4:    120 ....
    5:     45 .....
    6:     15 ......
    7:      1 .......


grep -v "[aeiou]" just_heterograms.txt | grep "^.......$" | tr '\n' ' ' ; echo
    glyndwr 

grep -v "[aeiou]" just_heterograms.txt | grep "^......$" | tr '\n' ' ' ; echo
    bsdhyg bsfmgt crwths crypts flysch ftncmd glyphs kfrsch khlyst lymphs nymphs schftz schwyz strych wrycht 

grep -v "[aeiou]" just_heterograms.txt | grep "^.....$" | tr '\n' ' ' ; echo
    blyth bsgph bshyg byrls chynd clwyd clywd crwth crypt cynth dryth fhlmc fldxt fyrds glyph gryph grypt gymps hwyls hymns hyrst kydst kynds lymph lynch mfnch mysql myths nymph psych rhynd rynds scyld scyth smyth syftn sylph synch synth thyms tryck tryms tryps tymps wynds
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