# As easy as Three, Two, One… How fast can you go from Five to Four?

You have to reach Four from Five in the minimum number of tries with the following rules.

• You are only allowed to change one letter at a time.
• You have to keep the word length the same.
• At each step of the way, you must have a legitimate English word. Thus, something like going from "FIVE" to "FOVE" is not a valid step, but from "BEAN" to "BEEN" would be.

For example: How do you go from $$LAND$$ to $$SANE$$?

Answer: $$LAND$$ $$\rightarrow$$ $$SAND$$ (1) $$\rightarrow$$ $$SANE$$ (2). Done in 2 steps, all real words.

Just signed up to share some of the solutions (3 of them) I was able to come up with in 6 steps:

1) FIVE $$\rightarrow$$ FAVE $$\rightarrow$$ FARE $$\rightarrow$$ FART $$\rightarrow$$ FAUT $$\rightarrow$$ FAUR $$\rightarrow$$ FOUR
2) FIVE $$\rightarrow$$ FAVE $$\rightarrow$$ FACE $$\rightarrow$$ FACT $$\rightarrow$$ FAUT $$\rightarrow$$ FAUR $$\rightarrow$$ FOUR
3) FIVE $$\rightarrow$$ DIVE $$\rightarrow$$ DOVE $$\rightarrow$$ DORE $$\rightarrow$$ DORR $$\rightarrow$$ DOUR $$\rightarrow$$ FOUR

OH MAN! After a lot of searching I was able to do it in 5 steps (5 solutions):

1) .. FIVE $$\rightarrow$$ FIRE $$\rightarrow$$ ... FORE .... $$\rightarrow$$ FORD $$\rightarrow$$ FOUD $$\rightarrow$$ FOUR
2,3) FIVE $$\rightarrow$$ FINE $$\rightarrow$$ FIND/FONE $$\rightarrow$$ FOND $$\rightarrow$$ FOUD $$\rightarrow$$ FOUR
4,5) FIVE $$\rightarrow$$ FINE $$\rightarrow$$ FINS/FONE $$\rightarrow$$ FONS $$\rightarrow$$ FOUS $$\rightarrow$$ FOUR

I confirmed the words on Anagrammer.

EDIT: although all words are accepted on Anagrammer, someone pointed out in the comments that two of the words used in the 6-step solutions should be considered French. I agree with that, so I am not sure about the correctness of those answers. However, the 5-step solutions use only english words.

• Nice! Welcome to the site, and +1. – Rand al'Thor Jun 9 '19 at 17:43
• @JoãoBravo Ah, I didn't spot that. Apologies. – Rand al'Thor Jun 10 '19 at 17:32
• Note that "foud" is an extremely obscure word (it means a bailiff, sheriff, magistrate, or governor in the Shetland & Orkney Islands), and "fous" hasn't been used regularly since Middle English according to the OED. – Michael Seifert Jun 11 '19 at 13:16
• @MichaelSeifert "fone" (plural of foe), "fons" (fool), and "lins" (to desist) are also obsolete from Middle English, while "faut" and "dore" are French with "faur" is from Scots (not Scottish, Scots). It feels like playing really fast and loose with the rules. – Draco18s no longer trusts SE Jun 11 '19 at 18:20
• @JoãoBravo "faut" and "dore" are French, not English, and those are valid. See my point? – Draco18s no longer trusts SE Jun 11 '19 at 18:38

I guess I am just not very good at these puzzles... the solution I found is 3,001 steps long without repeating any words.

Pardon any profane words that may be in this list. It was generated using the official TWL06 dictionary of Scrabble-legal words. Also note that I make no claim that this is the longest possible solution that does not repeat any words. It does seems to be of a rare variety though.

• The question asks for the shortest, not longest, ladder. – Herb Jun 11 '19 at 3:19
• Yeah, well he tried ;) – Cullub Jun 11 '19 at 14:48
• In some ways this is just as impressive as the shortest sequence – Roby Vicary Jun 13 '19 at 7:40
• Wow, you're remarkably bad at this. :) – Rubio Jun 13 '19 at 10:35

In this answer I will show that five steps is the minimum number required (from the answer by @JoãoBravo).

Suppose by contradiction that there are four. Then the sequence will be of the form

F i v e, _ _ _ _, _ _ _ _, _ _ _ _, F o u r.

If the first letter remains unchanged the whole way through, the sequence is

F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r.

For this case, please see the answer by @TimPillards, as the previous part of my answer has been demonstrated to be incorrect.

Now suppose that the first letter is changed to a letter ¬ the whole way through. Then the sequence is

F i v e, ¬ _ _ _, ¬ _ _ _, ¬ _ _ _, F o u r.

However, changing from i v e to o u r requires three steps, which are not allowed in this arrangement as the first and fourth steps are wasted.

Thus the only combinations left are

1. F i v e, F _ _ _, ¬ _ _ _, ¬ _ _ _, F o u r.

This is also impossible as there are not enough steps (one) to change from i v e to o u r.

1. F i v e, ¬ _ _ _, F _ _ _, F _ _ _, F o u r.

Steps two and three are pointless/a waste of steps so this case can be easily discarded.

1. F i v e, F _ _ _, F _ _ _, ¬ _ _ _, F o u r.

Yet again, steps three and four are useless for the same reason above.

1. F i v e, ¬ _ _ _, ¬ _ _ _, F _ _ _, F o u r.

There are not enough steps to make the transition i v e to o u r as the third step is wasted in converting ¬ to F.

1. F i v e, F _ _ _, ¬ _ _ _, F _ _ _, F o u r.

Again, not sufficient due to the same reasoning above.

1. F i v e, ¬ _ _ _, F _ _ _, ¬ _ _ _, F o u r.

This is also useless.

• Yes..for this case, it is true...my upcoming puzzle will lend to lot of interesting analysis like this. – Uvc Jun 9 '19 at 18:31
• You should state that any "waste of steps" means that you have less than the required 3 steps to make the transition. This would allow you to consolidate all the expanded examples to "You cannot change the F, because we only have 4 transitions (3 are required by Hamming distance, 1 spare), and changing F to ¬ to F takes 2 spares." – Cireo Jun 10 '19 at 23:34
• This proof is obviously wrong. Five=>Fove=>Foue=>Four is a perfectly fine 3 transition (sure those aren't words, but the proof doesn't rely on word lists). What (I think) is true is that if you can do it in 4 steps you can also do it in 3 and if you can't do it in 3, the next possible options is 5. – Voo Jun 13 '19 at 15:16
• @Voo Please see the answer below for the details which I originally overlooked when writing the answer. – TheSimpliFire Jun 13 '19 at 18:33
• @TheSimpliFire Originally? The first sentence is bold and says "I will show that five steps is the minimum number required". Which given my counter example is obviously wrong. – Voo Jun 13 '19 at 18:48

As @RossMillikan pointed out correctly, the proof from @TheSimpliFire is incomplete.

The following part

F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r. Therefore, at each step, one of i v e must be changed to match o u r, without any deviations. This is clearly impossible.

can be done with any of the following sequences
Five => F_ve => F_ue / F_vr => F_ur => Four
Five => Fi_e => Fo_e / Fi_r => Fo_r => Four
Five => Fiv_ => Fov_ / Fiu_ => Fou_ => Four

The first and last can easily be ruled out which leaves the second possiblity. I checked all 26 letters on Anagrammer and the furthest we get is with R as Fire and Fore do exist but we get blocked on Forr.

Which proofs that it is indeed not possible to do it with less than five steps and that @JoãoBravo found the shortest route from Five to Four.

• FORR does exist (en.wikipedia.org/wiki/FORR), but it's unfortunately an abbreviature, not a word, and this is not allowed. – trolley813 Jun 11 '19 at 6:46

I can do it in

7 steps: five fire fore fort port pout pour four.

With one more obscure word, I can do it in

6 steps: five fire tire tore torr tour four.

• Absolute minimum is obviously three..I wrote down in 7 without optimization..it will be fascinating to see if somebody can come up with a 5 step solution to better yours. – Uvc Jun 9 '19 at 14:33
• Three is definitely not possible because none of FOVE, FIUE, FIVR is a word. – Gareth McCaughan Jun 9 '19 at 14:54
• True..I just made a general statement regarding the minimum steps needed..in the example I cited, was able to do in 2 because middle letters were same. In general, absolute minimum would be..total letters - number of common letters in starting and ending words – Uvc Jun 9 '19 at 14:59
• Ah dang it. I had Live, Love, Lose, Lost, Lout, Tout, Tour, Four - 8 steps – Stian Yttervik Jun 10 '19 at 21:18

Four, but using more ancient spelling (where U and V were interchangeable)

fiue (five) - flue - flur - four

I found a solution in seven steps.

FIVE - FILE
FILE - FILL
FILL - FALL
FALL - FAIL
FAIL - FOIL
FOIL - FOUL
FOUL - FOUR

• Ha! Ninjaed you by nine seconds (though, oops, I wrote mine down in reverse order, which I'll fix). – Gareth McCaughan Jun 9 '19 at 14:23
• It is fascinating to see how each mind works through intermediate steps...I went through “Fire” to “Four”. – Uvc Jun 9 '19 at 14:35
• This is the solution I found - which is how I know that this one only uses words from my /usr/share/dict/words (in my distro, at least) – Alex F Jun 13 '19 at 5:00

Seven steps (with some obscurer words):

FOUR
LOUR
LOUK
LOCK
LICK
LICE
FICE
FIVE

Something that might lead to five, if it can be completed:

FIVE
FIRE
FIRM
FORM

That's three steps from FIVE to something that shares two letters with FOUR.

• And I see others have beaten me to seven even without obscure words ... – Rand al'Thor Jun 9 '19 at 14:34
• Interesting..in mine..F was never changed..still Ineeded seven..it would be really great if we can find 5 step solution at least. – Uvc Jun 9 '19 at 14:38
• To make your partial thing into a five you'd need either FOUM or FORR to be a word, and I think neither is one. – Gareth McCaughan Jun 9 '19 at 14:55