How Many Shiftgrams in a Word-Set?

Word Ways, February 2010 | Go to article overview

How Many Shiftgrams in a Word-Set?


A word possesses a shiftgram if its letters can first be shifted along the alphabet and then transposed to form another word. The classic example MUSIC to QUACK was offered by Alice Gorki (she is also a shiftgram) and Dimitri Miller in "Sea-Changed Words" in the February 1969 Word Ways. (The article was very likely written by Dmitri Borgmann and revised by Howard Bergerson after Dmitri resigned the Word Ways editorship.)

MUSIC-NVTJD-OWUKE-PXVLF-QYWMG-RZXNH-SAYOI-TEZPJ-UCAQK to QUACK

In 'Shiftgrams Revisited" in the November 2002 Word Ways, Darryl Francis asked "Is there some way of predicting an approximate number of shiftgrams that can be produced from a given set of words?" The number of shiftgrams obviously increases with the size of the word set, but it appears unlikely that a simple predictive formula can be devised.

To see why, it is useful to simplify the problem slightly. Any set of words (say, Webster's Unabridged) can be first edited by removing all but one example of each transposal group, and then sorted into shiffgroups of one to twenty-six words in size. In a shiftgroup of size one, the sole word has no shiftgrarn, so Darryl's question reduces to ascertaining the number of words in shiftgroups of size two or more. We illustrate this process for the two-letter words in the Official Scrabble Players Dictionary (OSPD); the 85 isograms are reduced to the following 68 in 13 groups, with an average of 5.23 per group:

 1 ab de ef hi no op ut 2 ay om us 3 ad ax be by he if li lo or
xu 4 ae aw mi so 5 fa id in jo to up 6 ye 7 ah at bi el ex ho pi nu 8 ai
as em we go mu ow 9 ar en of ox 10 ka oe is oy 11 la pa do pe et sh it
xi 12 am es my 13 an boer 

Similarly, the 585 three-letter isograms in the OSPD after duplicate transposals have been removed can be sorted into the following 100 shiftgroups, with an average of 5.85 per group:

 1-1 cab fed ghi nora uts 1-2 bad ged fig khi tool pro 1-3 adz
wax nor tux edh fie sop 1-4 vaw him son die jin top 1-5 upo bag vau sty
kef bin not 1-6 bah tau zag elf 1-7 mun von pow sat led but hog hip 1-8
now pox lum jab cut fen fog ars 1-9 kab foe den 1-10 alb doe his yon
1-11 cod ped vug hit set 1-12 ban cob bop refers tug 2-1 bay bed cad ilk
mop rut jig zax 2-2 ace cay gie way 2-3 day rid pur mor ink 2-4 aye keg
ice mig oms 2-5 fay try vat bid leg gin kip tom pun 2-6 sau gem yew 2-7
hay art vex sub wud eng foh pig irk pry 2-8 ego ski mow 2-9 lac par jay
peg vet rig why kit 2-10 cam yak yom sue 2-11 can nap lay bod ret erg
git 2-12 oca moa may 3-1 hie mil lop ors 3-2 fad dig beg kif nil rot pus
3-3 bey dag fil oil our 3-4 had axe lip taw sol 3-5 ump aid vas fax hem
fin lot row jog 3-6 hen spy 3-7 lis dak bel hoe rub 3-8 lad cur sap dog
wet hep lit low 3-9 mad neb oar urd fir wiz lox 3-10 ivy and rob obe her
rue ifs 3-11 ado lax map cor she fit fur jug 4-1 wab hid sot zed nim kop
put 4-2 age caw sou 4-3 daw fib hae buy tax lie imp kor run 4-4 awe saw
kos sow 4-5 rim pul sox 4-6 kea qua ism soy icy wag 4-7 ale hod apt pie
jun haw 4-8 mae cog guy wok sew 4-9 nae fob jaw sob ran ire vim rev ply
4-10 cos sei 4-11 pea alp pet law sod tie mix 5-1 fag dye chi fez ion
out 5-2 wry big sun kid nip ohm 5-3 sax lid web log two rum 5-4 ave dim
raw rin yup 5-5 din yid joe jot sin toy 5-6 tin syn yet 5-7 oat dip pub
sex 5-8 vin jow fan via bog bot cup hum 5-9 thy win oaf ann lug cot rid
pud 5-10 ids nix jet joy dot 6-1 tun hag fey poi dex elk noh 6-2 say gey
mog sum 6-3 jag bys rax hey fiz ken 6-4 eau oke you 6-5 kep 6-6 key yes
nth 6-7 ant lye hob rex pic hun hag 6-8 mug mas 6-9 lar flu yen shy duo
6-10 auk sic cwm uke 7-1 hop vow bat jib cub wed nog elm turn 7-2 cat
bud haj zig ray rip vox 7-3 tad hex lib ole rho sip 7-4 ate fub paw mib
ohs cry tip 7-5 hot piu fat ham bin ova bug dry 7-6 hub bio kex rye fly
sly tag 7-7 gun hat mat fry bun sel lex 7-8 hap ait how sal let pew pix
7-9 taj ark bir dun leu voe yip 8-1 sab dev cud few fon men tsk pyx 8-2
sac yow cue 8-3 sad ail pax bet hew old fix gor 8-4 sea aim pht 8-5 got
hup rem fas van ainjob bow jew 8-6 sag cow amu ems gym sky 8-7 fun has
alt pia bit bum met 8-8 ais ask cure emu mew owe goy 8-9 air jar fox mud
wen gox gyp hyp 9-1 bar due fop one oxy sir 9-2 arc fud pay veg pen mod
9-3 rad col jam dug for lux 9-4 are box wan vie em 9-5 far cox dui oft
sen pug 9-6 tic ten yep fou 9-7 rah kat dow lev sib hoy zip 10-1 jut zap
ted sit don feu ope 10-2 see wig 10-3 reb boy lob hue vis ore 10-4 oak
maw wis coy ose yuk 10-5 dit yod jeu toe yin six 11-1 fug tui azo des
con pod eft aim 11-2 cap any mob rec get per 11-3 pad rod coz the rug
pes wit 12-1 boa man nob red cop efs hug hut 

And finally we show the 25 shiftgroups into which the 80 three-letter words with two letters the same and one different can be sorted; here the average group size is only 3. …

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