Given an input string consisting of only letters, return the step-size that results in the minimum amount of steps that are needed to visit all the letters in order over a wrapping alphabet, starting at any letter.

For example, take the word, dog. If we use a step-size of 1, we end up with:

defghijklmnopqrstuvwxyzabcdefg   Alphabet

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

defghijklmnopqrstuvwxyzabcdefg   Visited letters

d          o                 g   Needed letters

For a total of 30 steps.

However, if we use a step-size of 11, we get:

defghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyzabcdefg

^          ^          ^          ^          ^          ^

d          o          z          k          v          g   Visited letters

d          o                                           g   Needed letters

For a total of 6 steps. This is the minimum amount of steps, so the return result for dog is the step-size; 11.

Test cases:

"dog"      -> 11

"age"      -> 6

"apple"    -> 19

"alphabet" -> 9

"aaaaaaa"  -> 0 for 0 indexed, 26 for 1 indexed

"abcdefga" -> 1 or 9

"aba"      -> Any odd number except for 13

"ppcg"     -> 15

"codegolf" -> 15

"testcase" -> 9

"z"        -> Any number

"joking"   -> 19

Rules

• Input will be a non-empty string consisting only of the letters a to z (you can choose between uppercase or lowercase)


• Output can be 0 indexed (i.e. the range 0-25) or 1 indexed (1-26)


• If there's a tie, you can output any step-size or all of them


• This is code-golf, so the lowest amount of bytes for each language wins!

Python 2, 222 bytes

A=map(chr,range(65,91))

def t(s,l,S=0):

 a=i=A.index(s[0]) 

 for c in s[1:]:

	while a!=A.index(c):

	 S+=1;a+=l;a%=26

	 if a==i:return float('inf')

	i=a

 return S

def f(s):T=[t(s,l)for l in range(26)];return T.index(min(T))

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-8 bytes from Jo King

This would be shorter in a language with a prime number of letters (wouldn't need the float('inf') handling of infinite loops). Actually, this submission would still need that for handling strings like "aaa".

This submission is 0-indexed (returns values from 0 to 25 inclusive).

f takes a(n uppercase) string and returns the Optimal Alphabet Stepping

t is a helper function that takes the string and an alphabet stepping and returns the number of hops needed to finish the string (or float('inf') if impossible).

JavaScript, 153 bytes

w=>[...Array(26)].map((_,i)=>i).map((_,s,a)=>~[...w].map(c=>(t+=a.find(v=>!p|(u(c,36)+~v*s-u(p,36))%26==0),p=c),p=t=0,u=parseInt)+t<m&&(m=t,n=s),m=1/0)|n

Try it online!