Number wheel Challenge!

Question

Much like the Word-tank-wheel challenge, but numbers!

Find a number that can 'rotate' in a way that the origin number and its rotations must be divisible (i.e. whole number) by the number of times it rotated.

Here's some examples of potential answer:

Example 1:
Number to start with: 321
Rotations: 3
Result:

origin number - 321 (321/3 = 107, pass)  
rotation 1: 213 (213/3 = 71, pass)  
rotation 2: 132 (132/3 = 44, pass)  
rotation 3: 321 (pass)  

Example 2:
Number to start with: 1644444
Rotations: 4
Result:

origin number - 1644444 (1644444 /4 = 411111, pass)  
rotation 1: 4164444 (4164444/4 = 1041111, pass)  
rotation 2: 4416444 (4416444/4 = 1104111, pass)  
rotation 3: 4441644 (4441644/4 = 1110411, pass) 
rotation 4: 4444164 (4444164/4 = 1111041, pass) 

Rules:
- Must be at least 3 digits & 3 rotations
- Rotation must be in the same direction
- Rotation ≤ Digit (to prevent infinite looping, see example 1)
- Cannot use repeating numbers (11111, 22222, etc) or number starting / ending with 0's (100, 01230, etc)
- Base 10 number system, positive numbers

Scoring:
- ([Digit] x [Rotations]) / (1 + (Digit - Rotation)), hence:
- Example 1 = 3 (Digits) x 3 (Rotations) / 1 = score of 9
- Example 2 = 7 (Digits) x 4 (Rotations) / 4 = score of 7

Highest score wins.
This challenge will remains open for at least 2 days.

Time's up!

Highest score: Jonathan Allan with score of ..um... ∞!
Great attempts with others who joined the challenge!

Answer 1

We can make a score of

...as high as we want* (subject to paper, ink and time - or "memory limitations")

* Update to method

An issue with the method given below is that $9$s become more frequent in $9^p$ as $p$ increases and we need to mutate more than just $2$ digits here and there. We can definitely do $p=28$ as $9^{28}$ has no $9$s, for a score of $72057594037927936$

Furthermore we can probably do this new method for any $p$ (not proven to myself yet):

Make $x=10^{9^{^{p}}}-1$
- a string of $9^p$ $9s$ (which will be divisible by $9^p$)
then make $n$ by subtracting a multiple of $9^p$ which has no $9$s but is less than $9^p$ digits long (actually we can go a bit higher if we really need to as the first digit of $x$ is a $9$)

As an example take $p=125$ - the smallest multiple of $9^p$ we can subtract in this case is $45409$, but that has only $124$ digits, nowhere near the $9^{125}-1$ digit limit we set ourselves above.

That would be $n=10^{9^{125}}-45409\times9^{125}-1$
obviously too big to write in an answer, and would give a score of $9^{250}$:

36360291795869936842385267079543319118023385026001623040346035832580600191583895484198508262979388783308179702534403855752855931517013066142992430916562025780021771247847643450125342836565813209972590371590152578728008385990139795377610001

Previous method

Make a string of $9$s of length $9^p-2$, place $2$ digits that sum to $9$ (e.g. $18$) somewhere in the string to make a value, $x$, of length $9^p$. The choice of where to place the two non-$9$ digits is made such that $n=x-(x\pmod{9^p})$ contains no zeros (this new string, $n$, is not going to be just $9$s and it will still have length $9^p$).

Such answers

quickly become too long to perform string manipulation in memory on my laptop (I could probably do something clever here I guess) but also would take an age to exhaustively check all rotations, although it is guaranteed due to the string's digits summing to a multiple of $9$ (we made our new $n$ congruent to $0$ modulo $9^p$) while being a length of $9^p$.

Here is an example

for which I have exhaustively tested all rotations, but which is too long for a SE answer box
using a $p$ of just $5$:
$18999\cdots99959256$

that is $18$ followed by $59042$ $9$s followed by $59256$
for a fully rotatable and divisible string of length of $9^5=59049$

For a score of

$\frac{59049^2}1=3486784401$

---

Original approach assumed "no repeated digits" meant something else and follows...

See further down for a score of

$43046721$

Pretty sure this can be extended for a higher score...

Score of $\frac{81\times81}1=6561$
$812345687923456789134567891245678912356789123467891234578912345689123456791234396$

Test in Python:

>>> n = 812345687923456789134567891245678912356789123467891234578912345689123456791234396
>>> s = str(n)
>>> for rotation in range(1, 82):
...     s = s[-1] + s[:-1]
...     if int(s) % 81:
...             print('failed at rotation {0}'.format(rotation))
... else:
...     print('success!')
...
success!
>>>

OK lets extend it - I use permutations of digits here to avoid repetition. More Python code:

>>>from itertools import permutations
>>> def makeN(l):
...     if l > 362880:
...             raise ValueError('Not enough permutations of 9 digits to satisfy the no repetition rule')
...     res = ''
...     digits = ''.join([str(i) for i in range(9, 0, -1)])
...     for p in permutations(digits):
...             for c in p:
...                     res += c
...                     if len(res) == l:
...                             res = int(res)
...                             res = res - (res % l)
...                             if '0' in str(res):
...                                     if res % 10 == 0 and l % 10 == 0:
...                                             raise ValueError('cannot produce a valid n for this l')
...                                     res += l
...                                     while '0' in str(res):
...                                             res += l
...                             return res
...

Now we can certainly score

$\frac{6561^2}1=43046721$ with makeN(6561)

With:

98765432198765431298765423198765421398765413298765412398765342198765341298765324198765321498765314298765312498765243198765241398765234198765231498765214398765213498765143298765142398765134298765132498765124398765123498764532198764531298764523198764521398764513298764512398764352198764351298764325198764321598764315298764312598764253198764251398764235198764231598764215398764213598764153298764152398764135298764132598764125398764123598763542198763541298763524198763521498763514298763512498763452198763451298763425198763421598763415298763412598763254198763251498763245198763241598763215498763214598763154298763152498763145298763142598763125498763124598762543198762541398762534198762531498762514398762513498762453198762451398762435198762431598762415398762413598762354198762351498762345198762341598762315498762314598762154398762153498762145398762143598762135498762134598761543298761542398761534298761532498761524398761523498761453298761452398761435298761432598761425398761423598761354298761352498761345298761342598761325498761324598761254398761253498761245398761243598761235498761234598756432198756431298756423198756421398756413298756412398756342198756341298756324198756321498756314298756312498756243198756241398756234198756231498756214398756213498756143298756142398756134298756132498756124398756123498754632198754631298754623198754621398754613298754612398754362198754361298754326198754321698754316298754312698754263198754261398754236198754231698754216398754213698754163298754162398754136298754132698754126398754123698753642198753641298753624198753621498753614298753612498753462198753461298753426198753421698753416298753412698753264198753261498753246198753241698753216498753214698753164298753162498753146298753142698753126498753124698752643198752641398752634198752631498752614398752613498752463198752461398752436198752431698752416398752413698752364198752361498752346198752341698752316498752314698752164398752163498752146398752143698752136498752134698751643298751642398751634298751632498751624398751623498751463298751462398751436298751432698751426398751423698751364298751362498751346298751342698751326498751324698751264398751263498751246398751243698751236498751234698746532198746531298746523198746521398746513298746512398746352198746351298746325198746321598746315298746312598746253198746251398746235198746231598746215398746213598746153298746152398746135298746132598746125398746123598745632198745631298745623198745621398745613298745612398745362198745361298745326198745321698745316298745312698745263198745261398745236198745231698745216398745213698745163298745162398745136298745132698745126398745123698743652198743651298743625198743621598743615298743612598743562198743561298743526198743521698743516298743512698743265198743261598743256198743251698743216598743215698743165298743162598743156298743152698743126598743125698742653198742651398742635198742631598742615398742613598742563198742561398742536198742531698742516398742513698742365198742361598742356198742351698742316598742315698742165398742163598742156398742153698742136598742135698741653298741652398741635298741632598741625398741623598741563298741562398741536298741532698741526398741523698741365298741362598741356298741352698741326598741325698741265398741263598741256398741253698741236598741235698736542198736541298736524198736521498736514298736512498736452198736451298736425198736421598736415298736412598736254198736251498736245198736241598736215498736214598736154298736152498736145298736142598736125498736124598735642198735641298735624198735621498735614298735612498735462198735461298735426198735421698735416298735412698735264198735261498735246198735241698735216498735214698735164298735162498735146298735142698735126498735124698734652198734651298734625198734621598734615298734612598734562198734561298734526198734521698734516298734512698734265198734261598734256198734251698734216598734215698734165298734162598734156298734152698734126598734125698732654198732651498732645198732641598732615498732614598732564198732561498732546198732541698732516498732514698732465198732461598732456198732451698732416598732415698732165498732164598732156498732154698732146598732145698731654298731652498731645298731642598731625498731624598731564298731562498731546298731542698731526498731524698731465298731462598731456298731452698731426598731425698731265498731264598731256498731254698731246598731245698726543198726541398726534198726531498726514398726513498726453198726451398726435198726431598726415398726413598726354198726351498726345198726341598726315498726314598726154398726153498726145398726143598726135498726134598725643198725641398725634198725631498725614398725613498725463198725461398725436198725431698725416398725413698725364198725361498725346198725341698725316498725314698725164398725163498725146398725143698725136498725134698724653198724651398724635198724631598724615398724613598724563198724561398724536198724531698724516398724513698724365198724361598724356198724351698724316598724315698724165398724163598724156398724153698724136598724135698723654198723651498723645198723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Answer 2

Score:81

819999999 over 9 rotations

Answer 3

for rotations = digits = divisior the highest score I could find was find is score 164

222,222,222,444,444,444 is divisible by 18 over 18 rotations

because all permutations are even and divisible by nine

but then I reasised:

888,888,888,888,888,888,444,444,444,444,444,444

is divisible by 36 for all permutations scoring 1296