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NOTE: The rules have been updated to cause the nature of the puzzle to be as intended. I will upvote the answer which was given before the change, but I look for both alphabetic and math operations to be used to solve the puzzle.

There are the kinds of puzzles where you take a number and try through math to get it to another number, or start with letters, and try to get to another word through a series of valid words... but what about a combination of the two.

The rules:

                           (15 14  5)
Starting with the letters: _ o  n  e_

and changing only one letter at a time or converting the letters to numbers, apply one mathematical operation specified below, you must follow the path of:

one $\to$ two $\to$ three $\to$ four $\to$ five $\to$ six $\to$ seven $\to$ eight $\to$ nine $\to$ ten

Each result of a letter substitution must be a valid English word. When converting to numbers, the results of the math do not need to be valid words when applying operations, but must be to switch back to letters (which must happen for each of the goalpoints).

Valid math operations: (all except rotation of numbers to a single characters digits)

$+1$,$-1$,$\times2$,$\mod27$,sum the digits (14 $\to$ 5),multiply the digits (14 $\to$ 4) rotate numbers clockwise or counterclockwise (15, 14, 5 $\to$ 14, 5, 15 etc)

Valid alphabetic operations:

add a letter, remove a letter, rotate characters clockwise or counterclockwise.

I will allow for roughly a month if there is interest and will accept the shortest path as the answer, with upvotes for partial solutions. Good luck!

Note: The switching of characters between numbers and letters does NOT count as a step, so does not penalize you.

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  • $\begingroup$ The rotation of numbers cannot be "to a single character's digits". $\endgroup$
    – Jonathan Allan
    Jun 10 '17 at 21:31
  • $\begingroup$ ...so are the others applied to each and every character or to a single one of our choosing (or either)? $\endgroup$
    – Jonathan Allan
    Jun 10 '17 at 21:32
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    $\begingroup$ Doesn't the "change a single letter" operation make all of the maths operations redundant? (as well as the rot13 single letter) $\endgroup$
    – Jaap Scherphuis
    Jun 10 '17 at 21:33
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    $\begingroup$ @JaapScherphuis Yep, I think the minimum path currently is dictated by Levenshtein distance. Edit: except words must me in a dictionary... $\endgroup$
    – Jonathan Allan
    Jun 10 '17 at 21:37
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    $\begingroup$ Updated wording to clarify what I meant by "step"... I meant each of one,two,three etc. $\endgroup$
    – ben-Nabiy Derush
    Jun 11 '17 at 13:32
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Note: This is currently invalid due to a change in the specification, but may give an aim for people.

Using only character substitution (now banned), character addition, and character removal (all words are currently linked to their definition in the Oxford English Dictionary):

42 steps (33 intermediaries):

one -> owe -> we -> wo -> two -> tho -> the -> thee -> three -> thee -> the -> toe -> foe -> fou -> four -> fou -> foe -> fie -> five -> fine -> sine -> sin -> six -> sin -> sen -> seen -> seven -> seen-> sen -> sin-> sign -> sigh -> sight -> eight -> sight -> sigh -> sign -> sin -> sine -> nine -> tine -> tin -> ten

Could the lack of restriction to dictionary words for "mathematical" operations help us...?

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  • $\begingroup$ Thank you for the initial answer. I updated the rules to reflect the desired goal of the puzzle. The idea is to use both math and alphabetic swaps to make the chain. I believe you will find that the math is now needed. This answer could serve as a jumping block though for possible word goals to shoot for in the switching to valid words. $\endgroup$
    – ben-Nabiy Derush
    Jun 11 '17 at 13:18

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