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In his novel A Case of Need, one of Michael Crichton's characters claims that the classic word‑ladder‡ "MOANS to ROCKY" can be made in 4 moves (5 lines total).
But the book does not prove this and the best solution it provides is 6 moves (7 lines).

I can do it in 5 moves (6 lines) as follows:

MOANS
MOONS
MOOKS
MOCKS
ROCKS
ROCKY

Can it be done in 4 moves?
If not, can that be proved?


‡ A classic word ladder transforms the starting word to the ending word, where each move changes only one letter at a time and each move consists of a valid dictionary word.
In the absence of a free online OED, here we will stipulate that the word must be in en.wiktionary.org and must have been there for at least a year.
Proper nouns are not allowed.

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3 Answers 3

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I'll attempt to prove it is not possible.

The words MOANS and ROCKY differ by four letters. To get a 4-step word ladder, no letter can be changed twice, and every letter must therefore change directly to its final value.

Therefore there are only 4 possibilities for the second word:
ROANS, MOCNS, MOAKS, MOANY
and four possibilities for the second to last word:
MOCKY, ROAKY, ROCNY, ROCKS

In the last set, MOCKY and ROAKY are not words, ROCNY is only a Slovak word, so we must use ROCKS.
In the first set, MOCNS is not a word, and MOAKS is discounted for being a proper noun, and we cannot use MOANY since our choice of ROCKS already changed the last letter. That leaves us with ROANS.

The ladder must therefore look like:
MOANS
ROANS
?????
ROCKS
ROCKY

The middle word can now only be ROCNS or ROAKS, neither of which are words.

EDIT:
If MOCKY is considered a word, then:

we could choose that for the second-to-last run in the ladder. Then ROANS can't be used for the first step, leaving MOANY as the only possibility there. This gives the ladder:

MOANS
MOANY
?????
MOCKY
ROCKY

The middle word must then be MOCNY or MOAKY, neither of which are words.

For completeness, here are all possible 4-step ladders from MOANS to ROCKY if you are not concerned whether all steps are actually words. There are $24=4!$ of them because there are $4$ changes that can be made in any order.

 1234    1243    1324    1342    1423    1432
 MOANS   MOANS   MOANS   MOANS   MOANS   MOANS
 ROANS   ROANS   ROANS   ROANS   ROANS   ROANS
 ROCNS   ROCNS   ROAKS   ROAKS   ROANY   ROANY
 ROCKS   ROCNY   ROCKS   ROAKY   ROCNY   ROAKY
 ROCKY   ROCKY   ROCKY   ROCKY   ROCKY   ROCKY
 
 2134    2143    2314    2341    2413    2431
 MOANS   MOANS   MOANS   MOANS   MOANS   MOANS
 MOCNS   MOCNS   MOCNS   MOCNS   MOCNS   MOCNS
 ROCNS   ROCNS   MOCKS   MOCKS   MOCNY   MOCNY
 ROCNY   ROCKS   ROCKS   MOCKY   ROCNY   MOCKY
 ROCKY   ROCKY   ROCKY   ROCKY   ROCKY   ROCKY
 
 3124    3142    3214    3241    3412    3421
 MOANS   MOANS   MOANS   MOANS   MOANS   MOANS
 MOAKS   MOAKS   MOAKS   MOAKS   MOAKS   MOAKS
 ROAKS   ROAKS   MOCKS   MOCKS   MOAKY   MOAKY
 ROCKS   ROAKY   ROCKS   MOCKY   ROAKY   MOCKY
 ROCKY   ROCKY   ROCKY   ROCKY   ROCKY   ROCKY
 
 4123    4132    4213    4231    4312    4321
 MOANS   MOANS   MOANS   MOANS   MOANS   MOANS
 MOANY   MOANY   MOANY   MOANY   MOANY   MOANY
 ROANY   ROANY   MOCNY   MOCNY   MOAKY   MOAKY
 ROCNY   ROAKY   ROCNY   MOCKY   ROAKY   MOCKY
 ROCKY   ROCKY   ROCKY   ROCKY   ROCKY   ROCKY

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  • $\begingroup$ I agree that "mocky" is not a word. But interestingly that same "Oxford Dictionary", that gnovice used, says it is. Albeit a bad, bad word. As with gnovice's words, my print OED agrees that "mocky" is not a word. Is that online "Oxford Dictionary" some kind of swotty Urban Dictionary? $\endgroup$
    – Brock Adams
    Commented Jul 21, 2018 at 8:43
  • $\begingroup$ Merriam-Webster has it, too. It's just a word that, thankfully, isn't used much. I guess the character in Mr Critchton's book isn't concerned about passing valid words in word ladders and only wants to say that you can go from MOANS to ROCKY in four steps, because the O is already there. $\endgroup$
    – M Oehm
    Commented Jul 21, 2018 at 15:05
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Here is one solution:

MOANS
MOAKS
MOCKS
ROCKS
ROCKY

However,

"Moaks" is only an entry because it is the plural of a surname. This satisfies the standard put forth in your question, but it's unlikely this is the intended solution, if one exists.

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  • $\begingroup$ Yeah, traditionally proper nouns are not allowed, but neither I nor the novel explicitly stated that; sorry. Also, I'd say that any name that is mostly less popular than "Gart" should not be allowed irregardless. ;) Correcting the question... $\endgroup$
    – Brock Adams
    Commented Jul 21, 2018 at 4:22
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I found one way to do it in four, if you allow some Medieval English:

MOANS
ROANS: Horses with a particular coat color pattern
ROANY: A type of thick tar
ROAKY: Hazy, nebulous, not clear or damp, foggy, misty
ROCKY

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    $\begingroup$ Kudos, but neither ROANY, nor ROAKY are in Wiktionary (ROANY, ROAKY). Ditto my Oxford English Dictionary (2008 edition); so this doesn't work. Also ROANY appears to be Scottish slang, last used circa 1809. There's almost no chance that Crichton referred to this. (¿Did some shenanigans make into that online OED page?) $\endgroup$
    – Brock Adams
    Commented Jul 21, 2018 at 7:26

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