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Reflected Halves (Posted on 2013-02-19) Difficulty: 3 of 5
The triple of words LIES, SEAM, MAIL has a property that the first half of each word is the reflection of the second half of the previous word in the list (wrapping around last to first at the end).
Are there more triples like this?
How long of a list can be constructed?

For five letter words DEVIL, LIMIT, TIMED comes close, with the V in DEVIL not matching the middle Ms in the other words.
Does a proper five letter triple exist?
What about six letter words (Halves of three letters)?

No Solution Yet Submitted by Brian Smith    
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Solution computer exploration | Comment 1 of 5

Programs basically like the following were used. Varied were the specific word lists used, the maximum length of chain sought, and whether or not plurals were allowed, with plural being determined by whether a word ending in S would still be a word if the S were removed.

DECLARE FUNCTION isWord# (w$)
DECLARE SUB addOn ()
CLEAR , , 25000
DEFDBL A-Z
DIM SHARED lst$(12), numb, size, hsize
size = 6
hsize = -INT(-size / 2)
w$ = SPACE$(size)
CLS
OPEN "\words\words" + LTRIM$(STR$(size)) + ".txt" FOR BINARY AS #1
OPEN "\words\words" + LTRIM$(STR$(size)) + ".txt" FOR BINARY AS #10
OPEN "reflected halves" + STR$(size) + ".txt" FOR OUTPUT AS #2
DO
    GET #1, , w$
    IF EOF(1) THEN EXIT DO
    good = 1
    IF RIGHT$(w$, 1) = "s" THEN
        IF isWord(LEFT$(w$, LEN(w$) - 1)) THEN good = 0
    END IF
    IF good THEN
        lst$(1) = w$: numb = 1
        PRINT w$
        addOn
    END IF
LOOP
CLOSE

SUB addOn
wd$ = SPACE$(size)
lt$ = ""
FOR i = size TO size - hsize + 1 STEP -1
    lt$ = lt$ + MID$(lst$(numb), i, 1)
NEXT
low = 1: high = LOF(10) / size
DO
    mid = INT((low + high) / 2)
    GET #10, (mid - 1) * size + 1, wd$
    IF LEFT$(wd$, hsize) > lt$ THEN high = mid - 1
    IF LEFT$(wd$, hsize) < lt$ THEN low = mid + 1
LOOP UNTIL high < low OR LEFT$(wd$, hsize) = lt$
IF LEFT$(wd$, hsize) = lt$ THEN
    WHILE LEFT$(wd$, hsize) = lt$ AND mid > 1
        mid = mid - 1
        GET #10, (mid - 1) * size + 1, wd$
    WEND
    IF LEFT$(wd$, hsize) <> lt$ THEN
        mid = mid + 1
        GET #10, (mid - 1) * size + 1, wd$
    END IF
    WHILE LEFT$(wd$, hsize) = lt$
        good = 1
        IF RIGHT$(wd$, 1) = "s" THEN
            IF isWord(LEFT$(wd$, LEN(wd$) - 1)) THEN good = 0
        END IF
        IF good THEN
            IF wd$ <> lst$(numb) THEN
                numb = numb + 1
                lst$(numb) = wd$
                IF numb > 3 AND wd$ = lst$(1) THEN
                    FOR i = 1 TO numb
                        PRINT #2, lst$(i); " ";
                    NEXT: PRINT #2,
                END IF
                IF numb < 6 AND wd$ > lst$(1) THEN
                    addOn
                END IF
                numb = numb - 1
            END IF
        END IF
        mid = mid + 1
        GET #10, (mid - 1) * size + 1, wd$
    WEND
END IF
END SUB

FUNCTION isWord (w$)
n = LEN(w$)
IF w$ = "i" THEN isWord = 1: EXIT FUNCTION
w1$ = SPACE$(n)
OPEN "\words\words" + LTRIM$(STR$(n)) + ".txt" FOR BINARY AS #2
l = LOF(2) / n
low = 1: high = l
DO
    mid = INT((low + high) / 2)
    GET #2, (mid - 1) * n + 1, w1$
    IF w1$ = w$ THEN isWord = 1: CLOSE 2: EXIT FUNCTION
    IF w1$ < w$ THEN low = mid + 1: ELSE high = mid - 1
LOOP UNTIL low > high
isWord = 0
CLOSE 2
END FUNCTION

For 4-letter words many results were found, so a shorter word list could be used, giving more common words. In fact, even with this, so many such rings are available that S plural endings were disallowed as mentioned above, so LIES, SEAM, MAIL, (LIES) does not appear. But if you really want a triple before the longer rings, KEEL, LEER, REEK, (KEEL) does appear below, as does NOES, SEEP, PEON, (NOES), with NOES evading the plural test by being an -ES plural rather than an -S plural. In each set, the initial word is repeated, so the loop is really one word shorter than each shown set:

abet teat talc clot toot tout tuba abet
abet teat talc clot tout tuba abet
abet teat tang gnat taut tuba abet
abet teat tart trot toot tout tuba abet
abet teat tart trot tout tuba abet
abet teat taut tuba abet
abet teem meek kelp plop pout tuba abet
amen neon noes seam mama amen
amen neon noes seer ream mama amen
amen neon nova avow wool loam mama amen
anal lair rink knot toot tout tuna anal
anal lair rink knot tout tuna anal
anal lame emir rink knot tout tuna anal
anal lard drab barb brat taut tuna anal
anal laud duet teat taut tuna anal
anal laud dung gnat taut tuna anal
anal laud dunk knot toot tout tuna anal
anal laud dunk knot tout tuna anal
anal lava avow wool loot tout tuna anal
anew weep peat talc clot tout tuna anew
anew weep peat tang gnat taut tuna anew
anew weep peat tart trot tout tuna anew
anew weep peat taut tuna anew
anew weep peek kelp plop pout tuna anew
anew weep peep peat taut tuna anew
anew weep peep pert trot tout tuna anew
anew weep pert trot toot tout tuna anew
anew weep pert trot tout tuna anew
anew weer ream mart trot tout tuna anew
anew weer reap pang gnat taut tuna anew
anew weer reek kelp plop pout tuna anew
anon noes seam mart trot tout tuna anon
anon noes seep peat taut tuna anon
anon noes seep peep peat taut tuna anon
anon noes seep pert trot tout tuna anon
anon nova avow wool loot tout tuna anon
anus surf fray yank knot tout tuna anus
anus surf fret teat taut tuna anus
aped deer reap papa aped
aped deer reed deer reap papa aped
aped deer reek keel leer reap papa aped
aped deer reel leer reap papa aped
avid dime emir rink knob boon nova avid
avow woof foal lava avow
avow woof fort trim mien neon nova avow
avow wool loam maim mien neon nova avow
avow wool loom monk knob boon nova avow
avow wool loon noes seep peal lava avow
avow wool loon noes seep peon nova avow
avow wool loon nova avow
barb brag garb brow woof ford drab barb
barb brew weep peal lard drab barb
barb brew weep peep peal lard drab barb
barb brow woof foal lard drab barb
barb brow woof ford drab barb
barb brow woof ford drab bard drab barb
boon noes seam maim mink knob boon
boon noes seam mart trim mink knob boon
boon noes seep peal lair rink knob boon
boon noes seep peal laud dunk knob boon
boon noes seep pert trim mink knob boon
boon noes seer ream maim mink knob boon
boon noes seer reef feud dunk knob boon
boor roam maid dime emir rink knob boor
boor roam maim mime emir rink knob boor
boor roam maim mink knob boor
boor roam mart trim mink knob boor
bung gnat talc clad daub bung
bung gnat tang gnat talc clad daub bung
bunk knit ting gnat talc clad daub bunk
bunk knot tong gnat talc clad daub bunk
burp prom moat talc clad daub burp
clad dank knit ting gnat talc clad
clad dank knot tong gnat talc clad
clad dank knot toot tong gnat talc clad
clad dart trot tong gnat talc clad
clad dart trot toot tong gnat talc clad
clam maim mink knit ting gnat talc clam
clam maim mink knot tong gnat talc clam
clam mart trot tong gnat talc clam
clam mart trot toot tong gnat talc clam
clap pang gnat talc clap
clap pang gnat tang gnat talc clap
claw wart trot tong gnat talc claw
claw wart trot toot tong gnat talc claw
clay yank knit ting gnat talc clay
clay yank knot tong gnat talc clay
clay yank knot toot tong gnat talc clay
clef feud duet teat talc clef
clef feud duet teat tang gnat talc clef
clef feud dung gnat talc clef
clef feud dung gnat tang gnat talc clef
clef feud dunk knit ting gnat talc clef
clef feud dunk knot tong gnat talc clef
clod doer reap pang gnat talc clod
clod doer reef feud duet teat talc clod
clod doer reef feud dung gnat talc clod
clot toed deer reap pang gnat talc clot
clot tong gnat talc clot
clot tong gnat tang gnat talc clot
clot toot tong gnat talc clot
clot toot tong gnat tang gnat talc clot
clot toot tout turf fret teat talc clot
clot tout turf fret teat talc clot
deer ream maim mike eked deer
deer ream maim mink knit tier reed deer
deer ream maim mink knot toed deer
deer ream maim mink knot toot toed deer
deer ream mart trim mike eked deer
deer ream mart trim mink knot toed deer
deer ream mart trod doer reed deer
deer ream mart trod dour rued deer
deer ream mart trot toed deer
deer ream mart trot toot toed deer
deer reap pail liar rake eked deer
deer reap pang gnat tart trot toed deer
deer reap pang gnaw wart trot toed deer
deer reef feud duel leer reed deer
deer reef feud duet teed deer
deer reef feud duke eked deer
deer reef feud dunk knit tier reed deer
deer reef feud dunk knot toed deer
deer reef feud dunk knot toot toed deer
deer reek keel leek keel leer reed deer
deer reek keel leer reed deer
deer reek keel leer reel leer reed deer
deer reek kelp plod doer reed deer
deer reek kelp plod dour rued deer
deer reel leek keel leer reed deer
deer reel leek kelp plod doer reed deer
deer reel leek kelp plod dour rued deer
deer reel leer ream maim mike eked deer
deer reel leer ream mart trot toed deer
deer reel leer reed deer
deer reel leer reef feud duet teed deer
deer reel leer reef feud duke eked deer
deer reel leer reek keel leer reed deer
deer reel leer reel leer reed deer
dime emir rink knit tier ream maid dime
doer ream mart trod doer
doer reap pang gnat tart trod doer
doer reap pang gnaw wart trod doer
doer reef feud duel leek kelp plod doer
doer reef feud duet teat tart trod doer
doer reef feud dung gnat tart trod doer
doer reef feud dung gnaw wart trod doer
doer reek keel leek kelp plod doer
doer reek keel leer ream mart trod doer
doer reek keel leer reek kelp plod doer
doer reek kelp plod doer
doer reel leek keel leek kelp plod doer
doer reel leek kelp plod doer
doer reel leer ream mart trod doer
doer reel leer reek kelp plod doer
doer reel leer reel leek kelp plod doer
duel leek keel leer reef feud duel
duel leer reef feud duel
duel leer reek keel leer reef feud duel
duel leer reel leer reef feud duel
duet teem meek keel leer reef feud duet
duet teen neon noes seep peal laud duet
duet teen neon noes seer reef feud duet
dunk knit tier reef feud dunk
dunk knit tier reel leer reef feud dunk
emir rink knit tier ream maim mime emir
emir rink knit tier reap pail lime emir
emir rink knit ting gnaw wail lime emir
emir rink knot toil lime emir
emir rink knot tong gnaw wail lime emir
emir rink knot toot toil lime emir
fray yank knit ting gnat taut turf fray
fray yank knot tong gnat taut turf fray
fray yank knot toot tout turf fray
fray yank knot tout turf fray
fret teat tang gnat taut turf fret
fret teat tart trot toot tout turf fret
fret teat tart trot tout turf fret
fret teat taut turf fret
fret teem meek kelp plop pout turf fret
gnat tang gnaw wart trot tong gnat
gnat tang gnaw wart trot toot tong gnat
gnat tart trim mink knit ting gnat
gnat tart trim mink knot tong gnat
gnat tart trim mink knot toot tong gnat
gnat tart trot tong gnat
gnat tart trot toot tong gnat
gnaw wart trim mink knit ting gnaw
gnaw wart trim mink knot tong gnaw
gnaw wart trim mink knot toot tong gnaw
gnaw wart trot tong gnaw
gnaw wart trot toot tong gnaw
keel leer reek keel
keel leer reel leek keel
keel leer reel leer reek keel
keel leer reel leer reel leek keel
keel leer reel leer reel leer reek keel
knit tier ream maim mink knit
knit tier ream mart trim mink knit
knit tier reel leer ream maim mink knit
maim mien neon noes seam maim
maim mien neon noes seer ream maim
mart trim mien neon noes seam mart
mart trim mien neon noes seer ream mart
mien neon noes seep peat tart trim mien
mien neon noes seep peep pert trim mien
mien neon noes seep pert trim mien
noes seep peep peon noes
noes seep peon noes

With 5-letter words, solutions were uncommon enough that I show below the smaller dictionary (i.e., more-common words), but with plurals allowed:

delis silos soled delis
delis silos solos soled delis
dimer remit timid dimer
keels sleek keens sneer reels sleek keels
keels sleet teens sneer reels sleek keels
keens sneer reels sleek keens
loons snort trots stool loons
loons snort trots stool loots stool loons
loons snots stool loons
loons snots stool loots stool loons
loons snots stool loots stool loots stool loons
reels sleet teens sneer reels

Only the third row, with a 3-element ring, has no plurals.

Using another word list downloaded from the Internet gives a better picture, without plurals, though oddly "dimer" is not present. Notice also the program doesn't check for belated repetition:

debut tubal labor robed debut
debut tuber rebut tubal labor robed debut
debut tuber rebut tubed debut
debut tuber rebut tuber rebut tubed debut
decaf faced decal laced decaf
decaf faced decal laced decal laced decaf
demur rumor roman named demur
denim minim mined denim
devil livid divan naval laved devil
devil livid dived devil
level lever revel level
level lever revel lever revel level
level lever revel lever revel lever revel level

7-letter words:

deified deifier reified deified
deified deifier reified deifier reified deified
deified deifier reifier reified deified
deifier reifier reified deifier
delimed demirep periled delimed
delimed demirep periled deliver reviled delimed
delimed demised desired derived deviled delimed
delimed demised desired deriver reviled delimed
deliver reviler relived deviled deliver
deliver reviler relived deviser resiled deliver
deliver revised desired derived deviled deliver
deliver revised desired deriver reviled deliver
deliver reviser resiled deliver
deliver revived deviled deliver
deliver revived deviser resiled deliver
deliver reviver reviled deliver
deliver reviver reviler relived deviled deliver
deliver reviver reviser resiled deliver
deliver reviver revived deviled deliver
deliver reviver revived deviser resiled deliver
derived devised desired derived
deriver reviler relived devised desired deriver
deriver revised desired deriver
deriver revived devised desired deriver
deriver reviver revised desired deriver
deriver reviver revived devised desired deriver

 


  Posted by Charlie on 2013-02-19 18:41:14
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