# Fastest way from 10 to 1 with everyone in between

I got inspired by this question from trolley813 to make this puzzle.

How fast one can get from TEN to ONE, passing each number in between?

The rules are simple:

• You are only allowed to change one letter at a time.
• At each step of the way, you must have a legitimate English word. Thus, something like going from "EIGHT" to "EIGHN" is not a valid step, but from "FIGHT" to "NIGHT" would be.
• Here is an example going from TEN to NINE: TEN - TENT - LENT - LINT - LINE - NINE

Good luck!

• Do you require the intermediate numbers to be passed in order or not? – Gareth McCaughan Jun 12 '19 at 10:41
• Yes, that was the intention – Pim Schwippert Jun 12 '19 at 10:43
• When going from SIX to FIVE should the extra letter be added first, or last? And when going from THREE to TWO should the letters be removed first or last? Must they be the end two letters? – Weather Vane Jun 12 '19 at 10:44
• That's up to you :) – Pim Schwippert Jun 12 '19 at 10:45
• That would have been my next puzzle to be posted today..just the reverse...I even got the title ready....it would have been..wow!!,,zero to sixty in 3 seconds..zero to ten not so fast...with some rule changes. – Uvc Jun 12 '19 at 10:47

Here's one way. It probably isn't close to optimal. 42 steps if I've counted right.

TEN tin tine NINE sine sinh sigh sight EIGHT sight sigh sign sin sen seen SEVEN seen see sex SIX sin sine fine FIVE fire fir for FOUR for foe toe the thee THREE tree tee toe to TWO to ton tone ONE

Credit where due: the path from EIGHT to SEVEN is derived from Hunter's answer to an earlier puzzle, though not much of that answer remains in what I have there now. JonMark Perry spotted what in hindsight should have been an obvious improvement in the path from FOUR to THREE.

Definitions of the least familiar words above:

sinh: hyperbolic sine; a Thai garment
sen: since; a Japanese coin
thew: strength

• you could go THE - THEE - THREE – JMP Jun 12 '19 at 11:20
• oh yes, so I could; how extremely stupid not to have thought of that. – Gareth McCaughan Jun 12 '19 at 11:20
• Because this puzzle is secretly nine puzzles wearing a trench coat, we can compare the number of steps taken to the Levenshtein distance between the two words; this should give a very useful estimate for choosing which intervals are likely to be the easiest ones to improve. As I'm writing this comment, the theoretically perfect intervals are TEN-NINE and SEVEN-SIX, and all the rest are one off the theoretical optimum, except EIGHT-SEVEN, which is is off by two. – Bass Jun 12 '19 at 11:43
• well done, that's awesome! – Pim Schwippert Jun 12 '19 at 12:29
• TWO->OWE is nothing like valid. Why would it be? – Gareth McCaughan Jun 13 '19 at 14:52

This is a proof that Omega Krypton's answer of 38 is optimal, as long as these words are nonexistent (single implication):

EEVEN,EINE,ENINE,FIUR,FIVR,FOVE,NIGE,NINH,OWO,,SEGEN,SEVET,SEVHN,SIVEN,TNO,TWE

First, a calculation of the theoretical best (although no suitable words may exist), by using a Levenshteins distance (word difference by letters) calculator:

NINE -> EIGHT = 4, Currently answers have 5 steps

EIGHT -> SEVEN = 5, Currently answers have 6+ steps

FIVE -> FOUR = 3, Currently answers have 4 steps

FOUR -> THREE = 5, Omega's answer optimized (there is some dispute there?)

TWO -> ONE = 3, Currently answers have 4 steps

Theoretical minimum = 3+4+5+4+3+3+5+4+3 = 34

Now let's analyse the algorithm and already existing answers on each non-optimized point to see if there is room for improvement:

NINE -> EIGHT = 4 This keeps the I intact, substitutes N,N,E, and adds T. One of the following words would have to exist: ENINE,EINE,NIGE,NINH.

EIGHT -> SEVEN = 5 This obviously is 5 substitutions. One of these would be needed(reverse from seven): EEVEN,SIVEN,SEGEN,SEVHN,SEVET

FIVE -> FOUR = 3 3 substitutions, would need: FOVE,FIUR,FIVR

TWO -> ONE = 3 3 substitutions, would need: OWO,TNO,TWE

Q.E.D.

• It's pretty easy to demonstrate that TWO to ONE isn't possible in 3 steps. To do that, your first step would need to change either the first, second or third letter of the source word (TWO) into the corresponding letter of the target word (ONE). That would give you OWO, TNO, or TWE, none of which are valid English words. – Anthony Grist Jun 13 '19 at 13:29
• @Anothony I didn't expect so many upvotes, so I took it a little further in much the way you describe :) – George Menoutis Jun 13 '19 at 13:39
• thank you so much! +1 – Omega Krypton Jun 13 '19 at 14:21
• What about en.wiktionary.org/wiki/segen? ;-) – SaiBot Jun 13 '19 at 14:48
• EINE and SEGEN are German words, if that helps ... – Hagen von Eitzen Jun 13 '19 at 15:40

## Best solution so far: 38 steps

9-7 from @Gareth, and sien from @devyndraen thanks!

TEN tin tine NINE sine sinh sigh sight EIGHT sight sigh sign sien seen SEVEN seen see sex SIX fix fie FIVE fie fir for FOUR for fore tore thre THREE thee the tho TWO to ton on ONE

• Couldn't find "thre" in any dictionary, tho. – Bass Jun 12 '19 at 13:04
• @Bass all words are supported by wiktionary, such as this: en.wiktionary.org/wiki/thre – Omega Krypton Jun 12 '19 at 13:10
• But it is obsolete... – justhalf Jun 13 '19 at 13:44
• @justhalf but they are english words... :) – Omega Krypton Jun 13 '19 at 14:21
• So far as I can tell SIEN is an obsolete spelling of SEEN and SCION and THRE an obsolete spelling of THREE. In my book, those aren't words, although they were words once. – Gareth McCaughan Jun 13 '19 at 14:54

39 steps (40 words):

TEN tin tine NINE sine sinh sigh sight EIGHT sight sigh sign sien seen SEVEN seen see sex SIX fix fie FIVE fire fore for FOUR for foe toe the thee THREE thee the tho TWO to ton tone ONE

Word 6 courtesy of @Gareth

Definition of less-known words:

sinh - hyperbolic sine; a Thai garment
sien - a scion; descendant (from Collin's English Dictionary)
tho - nonstandard spelling of though