Word Puzzle.

[stopmadnessnow]

Is it possible to create a three by three square of letters, with words in every direction (that is reading backwards and forwards) without repeating a letter? ARE/ERA seems an obvious entry, but SHE/EHS for instance might give more scope?

[jaap]

I don't think so.

Spoiler:

A quick google for "reversible words" led me to this page.
I then filtered out all the ones that are not 3 letters long, and removed eel/lee due to the repeated letter.
The remaining words are almost all consonant/vowel/consonant, except for:
are/era
ate/eta
bro/orb
So if you use the c/v/c words to form a square, you need two v/c/v words for the middle row/column, and there aren't enough of those assuming the list is complete. We need at least one other reversible v/c/v word where the vowels are not a or e, and the middle letter is r or t.

Other letter patterns (the c/c/v of bro or she) seem to be even more difficult to make work.

[Poker]

Yeah, with the word set jaap provided in the spoiler, it's not possible.

Spoiler:

Consider the word in the middle row. This word consists of the middle letter of the words in each of the three columns. Therefore, this word must be made entirely up of letters in the middle of some word. The only seven letters that appear in this way are A, E, I, O, R, T, and U. The only words in the list that are made up of those letters are are/era, ate/eta, rat/tar, and rot/tor. One of these four must be the word in the middle row.

By the same logic, one of those four must also be the word in the middle column, and must be a different word. However, no two of the four words above share a middle letter. So the middle row and middle column must be the same word, and so must share all their letters.

[stopmadnessnow]

Include foreign words?

[Demki]

In that case you need to explicitly define the dictionary you use. If you, for instance include Hebrew words, it is quite likely you'll be able to solve it since every letter can be a consonant.
It also opens up to something like chinese where each letter is a word in and of itself, and iirc order doesn't matter as much in chinese

[Sandor]

There is a Scrabble word list called SOWPODS that appears to be available on SourceForge (direct link to file). I have no idea if this list is the actual SOWPODS list, but I downloaded it anyway. It has 267,751 words in it, including 1292 3-letter words.

If you've got access to a Linux system, then here's one way to make a list of all reversable 3 letter words:

Code: Select all

# Extract 3 letter words from SOWPODS list
grep '^...$' sowpods.txt > 3lw.txt

# Reverse every line
cat 3lw.txt | rev > 3lw_rev.txt

# Find words that appear twice, so must be in both lists
cat 3lw.txt 3lw_rev.txt | sort | uniq -d > rev_3lw.txt


Here are all VCV words from that list (the 2nd half is first half reversed)

Code: Select all

grep '[aeiou][^aeiou][aeiou]' rev_3lw.txt | egrep -v '(.).\1'
abo
ado
are
ate
avo
era
eta
oba
oda
ova


All the VCC words

Code: Select all

grep '[aeiou][^aeiou][^aeiou]' rev_3lw.txt | egrep -v '.(.)\1'
ahs
arb
arf
ehs
ohm
orb
orf
owt
urb


All the CVV words

Code: Select all

grep '[^aeiou][aeiou][aeiou]' rev_3lw.txt | egrep -v '.(.)\1'
duo
kue
nae
ria
sae
tae
tao


Count of CVC words

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grep '[^aeiou][aeiou][^aeiou]' rev_3lw.txt | egrep -v '(.).\1' | wc -l
388


Letters in the middle of a word

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cat rev_3lw.txt | cut -b2 | sort -u | tr -d '\n' ; echo
abdeghiklmnortuvwxyz


Count of words made up of just middle letters

Code: Select all

mid="abdeghiklmnortuvwxyz"
grep "[$mid][$mid][$mid]" rev_3lw.txt | egrep -v '.(.)\1|(.).\2|(.)\3.' | wc -l
254


Can anybody solve it using this SOWPODS wordlist, or prove it can't be done?

[Poker]

Regarding SOWPODS, I picked up the words from here to use.

Spoiler:

We only need to consider word pairs - i.e. pairs of words such that each is the other reversed. Additionally, we only need to consider word pairs where the three letters used are each unique - so eel/lee, for example, would be eliminated. From these two properties, there are only 216 word pairs that need to be considered. Sandor's list of letters in the middle of words, it would seem, does not take into account the second property. When that property is taken into account, the letters in the middle of words are as follows: ABDEHIORTUVWY. From there, only 47 word pairs could fit in the middle. From these 47 words, only 6 letters are a part of multiple word pairs and so could be in the center of the square: AEIORU. Let us consider each of these.

Case I: The word pairs in this case are air/ria, bid/dib, dit/tid, and div/vid. Of the four, only air/ria does not contain the letter D, and so this must be one of the two word pairs, and the other must contain a D:

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?D?
AIR
???

However, the only word pair with a D in the center is ado/oda, which shares A with air/ria. Therefore, I cannot be the center letter.

Case R: The word pairs in this case are arb/bra, are/era, bro/orb, and bru/urb. Much like in case I, one word pair is are/era and the other must contain a B:

Code: Select all

?B?
ARE
???

However, the only word pair with a B in the center is abo/oba, which shares A with are/era. Therefore, R cannot be the center letter.

Case U: The word pairs in this case are bud/dob, bur/rub, but/tub, and duo/oud. As in the previous cases, one word pair is duo/oud and the other must contain a B:

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?B?
DUO
???

However, the only option for the top row, abo/oba, shares o with duo/oud. Therefore, U cannot be the center letter.

Case E: There are six word pairs in this case: bed/deb, dew/wed, her/reh, hey/yeh, tew/wet, and wey/yew. The pair bed/deb can immediately be eliminated, since b can only intersect abo/oba and d can only intersect ado/oda. Similarly, dew/wed can only intersect ado/oda and owt/two, and so can be eliminated immediately. Therefore, one word pair must be her/reh or hey/yeh, and the other must be tew/wet or wey/yew. W must intersect owt/two, and the only word pair that intersects H that has an O is mho/ohm:

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OHM
WE?
T??

The WE? must now be wey/yew, but the only word pair with y in the middle is nys/syn, which does not contain M. Therefore, E cannot be the center letter.

Case O: There are ten word pairs in this case. The letters BDW each can only intersect a word pair with an O in it (abo/oba, ado/oda, and owt/two, respectively), which eliminates every word pair except for rot/tor. Therefore, with only one word pair not immediately eliminated, O cannot be the center letter.

Case A: There are fourteen word pairs in this case. The letters BDTV can be eliminated since the word they intersect with must contain an A (abo/oba, ado/oda, ate/eta, and avo/ova, respectively). Eliminating those leaves four word pairs: hay/yah, raw/war, ray/yar, and way/yaw. The two word pairs used must be one with a Y, and raw/war:

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?Y?
RAW
???

However, W can only intersect owt/two, and Y can only intersect nys/syn. Those two word pairs share no letters, and so cannot simultaneously be put in place. Therefore, A cannot be the center letter.

As all six possibilities have been eliminated, no letter can be in the center, and so the square is impossible.