# Digital Cellular Automata

Write a program or function that takes in an odd positive integer N and a string of decimal digits (0123456789). The string represents a ten-state one-dimensional cellular automaton. Each digit occupies one cell and the update rule from one generation to the next is that every cell becomes the digit resulting from the sum of the N cells centered on the cell, modulo 10.

The first and last cells wrap around as if neighbors, so cells can always have N cells centered on them. Note that N may be larger than the length of the string, which means it could wrap around multiple times and some digits will accordingly be in the sum multiple times.

As an example, if N is 7 and the string is 038, to visualize the cells to sum we can write 038 repeating infinitely in both directions

...038038038038038...


then the digit that the 0 will change into is the sum of the 7 digits centered around any 0, modulo 10:

...038038038038038...
^_____^
|
sum all these


This is (0+3+8+0+3+8+0)%10, which is 2.

Similarly the digits the 3 and 8 change into are defined by (3+8+0+3+8+0+3)%10 = 5 and (8+0+3+8+0+3+8)%10 = 0 respectively.

Thus, the generation after 038 is 250 when N is 7.

Your program or function needs to print or return the digit string of the very next generation of the input digit string. i.e. apply the update rule once to each cell and give the output. The shortest code in bytes wins.

# Test Cases

[digit string] -> [N = 1], [N = 3], [N = 5], [N = 7], [N = 9], [N = 43]
0 -> 0, 0, 0, 0, 0, 0
1 -> 1, 3, 5, 7, 9, 3
2 -> 2, 6, 0, 4, 8, 6
3 -> 3, 9, 5, 1, 7, 9
4 -> 4, 2, 0, 8, 6, 2
5 -> 5, 5, 5, 5, 5, 5
6 -> 6, 8, 0, 2, 4, 8
7 -> 7, 1, 5, 9, 3, 1
8 -> 8, 4, 0, 6, 2, 4
9 -> 9, 7, 5, 3, 1, 7
00 -> 00, 00, 00, 00, 00, 00
07 -> 07, 47, 41, 81, 85, 47
10 -> 10, 12, 32, 34, 54, 12
11 -> 11, 33, 55, 77, 99, 33
12 -> 12, 54, 78, 10, 34, 54
34 -> 34, 10, 78, 54, 12, 10
66 -> 66, 88, 00, 22, 44, 88
80 -> 80, 86, 46, 42, 02, 86
038 -> 038, 111, 294, 250, 333, 472
101 -> 101, 222, 343, 545, 666, 989
987 -> 987, 444, 901, 765, 222, 543
1234 -> 1234, 7698, 3412, 9876, 1234, 7698
26697 -> 26697, 54128, 00000, 56982, 84413, 54128
001002 -> 001002, 211122, 331332, 335334, 455544, 113112
129577020 -> 129577020, 326194923, 474081605, 961120291, 333333333, 183342413
6023845292173530 -> 6023845292173530, 6853571632015189, 1197228291289874, 9238433109901549, 0110956118726779, 1982123699138828
• @LegionMammal978 Lets keep it as a string. – Calvin's Hobbies Nov 19 '15 at 18:02
• @LegionMammal978 No. I admit I could have allowed it originally but doing so now would unfairly affect existing answers that use strings. – Calvin's Hobbies Nov 19 '15 at 22:01
• Well, thanks for nearly doubling the size of my answer... – LegionMammal978 Nov 19 '15 at 22:06

# Pyth, 20 19 bytes

1 byte thanks to @FryAmTheEggman.

smesmv@z-+dk/Q2Qlz

• e<num> is equivalent to %<num>T – FryAmTheEggman Nov 19 '15 at 14:15

## CJam, 21 bytes

l~_,\2/f-l:~fm>:.+Af%


Test it here.

### Explanation

l~   e# Read and evaluate N.
_,   e# Duplicate and turn into range [0 1 ... N-1]
\2/  e# Swap with other copy and (integer) divide by 2.
f-   e# Subtract this from each element in the range to get
e# [-(N-1)/2 ... -1 0 1 ... (N-1)/2]
l:~  e# Read string and evaluate each digit separately.
fm>  e# Make one copy of the result for each element i in the range, shifting the array
e# i cells to the right, cyclically.
:.+  e# Sum the columns of the resulting matrix.
Af%  e# Take each of those sums modulo 10.


# Mathematica, 85 bytes

""<>ToString/@CellularAutomaton[{Tr@#~Mod~10&,{},#/2-1/2},FromDigits/@Characters@#2]&

• Can you use .5 instead of 1/2? – mbomb007 Nov 19 '15 at 21:08
• @mbomb007 No, it needs to be an integer. – LegionMammal978 Nov 19 '15 at 21:56

## Python 3, 1149286 80 bytes

Took off 6 bytes thanks to Sp3000 and another 6 bytes thanks to xnor!

a=lambda N,D,i=0:D[i:]and str(int((D*N)[(i-N//2)%len(D):][:N],11)%10)+a(N,D,i+1)


Defines a named function a that takes N and D as parameters, the N and digit string defined in the challenge.

### Explanation

In Python 3, and between two strings will end up being the latter. Hence, D[i:]and ... short-circuits once all center positions have been iterated over as D[i:] will be an empty string and therefore falsy. (D*N)[(i-N//2)%len(D):][:N] duplicates the digit string a bunch of times, then slices it in the right places to give the substring that has the correct digit as the center. Recall for a moment that the sum of the digits of a base 10 number modulo 9 is the same as the number itself modulo 9. str(int(...,10)%10) treats the resulting number-string as if it was base 11 and gets the remainder modulo 10, then converts back to string. Finally, a(N,D,i+1) moves on to the next center position. Because of the +, once the recursion is done, all of the resulting digits are lumped together and returned.

String conversion is really expensive in Haskell...

# NARS2000 APL, 37 characters (72 bytes)

⎕←10⊥10∣+⌿⊃({⍵..-⍵}⌊⎕÷2)∘.⌽⊂49-⍨⎕AV⍳⍞


Explanation:

  ⎕←10⊥10∣+⌿⊃({⍵..-⍵}⌊⎕÷2)∘.⌽⊂49-⍨⎕AV⍳⍞
⍝ ⎕←                                    output
⍝   10⊥                                 the base-10 digits in
⍝      10∣                              the modulo-10
⍝         +⌿                            column-wise sum of
⍝           ⊃                           the matrix version of
⍝                         ∘.⌽           the outer-product rotation of
⍝                            ⊂            the scalar version of
⍝                                 ⎕AV⍳    the index in the atomic vector of
⍝                                     ⍞   an input string
⍝                             49-⍨        minus 49 ('0' + 1)
⍝                                       by
⍝             {⍵..-⍵}                     the range ⍵ to -⍵, where ⍵ is
⍝                    ⌊                    the floor of
⍝                     ⎕                   an input integer
⍝                      ÷2                 divided by 2

• Isn't APL one byte per character, since the encoding isn't UTF-8? APL uses the APL code page. – mbomb007 Nov 19 '15 at 21:05
• @mbomb007 NARS2000 does not support the APL code page as far as I know, and the .. primitive is nonstandard and thus not "portable". – Oberon Nov 19 '15 at 21:43
• Would it maybe be shorter to use Dyalog APL? – mbomb007 Nov 19 '15 at 21:48

# Octave, 64 bytes

@(s,n)["" mod(sum(bsxfun(@shift,s'-48,(1:n)-ceil(n/2))'),10)+48]


# J, 41 bytes

"."0@]{~(1":10|+/@:)#@]|-:@<:@[-~(+/&i.#)


Turned out longer than I expected. Should be golfable.

We generate a matrix with elements in a row showing the positions whose values should be added (mod 10) to get sum for a position.

Usage:

   7 ("."0@]{~(1":10|+/@:)#@]|-:@<:@[-~(+/&i.#)) '038'
250
`

Try it online here.