## Challenge:

Input: A list of distinct positive integers within the range $$\[1, \text{list-size}]\$$.

Output: An integer: the amount of times the list is riffle-shuffled. For a list, this means the list is split in two halves, and these halves are interleaved (i.e. riffle-shuffling the list [1,2,3,4,5,6,7,8,9,10] once would result in [1,6,2,7,3,8,4,9,5,10], so for this challenge the input [1,6,2,7,3,8,4,9,5,10] would result in 1).

## Challenge rules:

• You can assume the list will only contain positive integers in the range $$\[1, \text{list-size}]\$$ (or $$\[0, \text{list-size}-1]\$$ if you choose to have 0-indexed input-lists).
• You can assume all input-lists will either be a valid riffle-shuffled list, or a sorted list which isn't shuffled (in which case the output is 0).
• You can assume the input-list will contain at least three values.

## Step-by-step example:

Input: [1,3,5,7,9,2,4,6,8]

Unshuffling it once becomes: [1,5,9,4,8,3,7,2,6], because every even 0-indexed item comes first [1, ,5, ,9, ,4, ,8], and then all odd 0-indexed items after that [ ,3, ,7, ,2, ,6, ].
The list isn't ordered yet, so we continue:

Unshuffling the list again becomes: [1,9,8,7,6,5,4,3,2]
Again becomes: [1,8,6,4,2,9,7,5,3]
Then: [1,6,2,7,3,8,4,9,5]
And finally: [1,2,3,4,5,6,7,8,9], which is an ordered list, so we're done unshuffling.

We unshuffled the original [1,3,5,7,9,2,4,6,8] five times to get to [1,2,3,4,5,6,7,8,9], so the output is 5 in this case.

## General rules:

• This is , so shortest answer in bytes wins.
Don't let code-golf languages discourage you from posting answers with non-codegolfing languages. Try to come up with an as short as possible answer for 'any' programming language.
• Standard rules apply for your answer with default I/O rules, so you are allowed to use STDIN/STDOUT, functions/method with the proper parameters and return-type, full programs. Your call.
• Default Loopholes are forbidden.

## Test cases:

Input                                                   Output

[1,2,3]                                                 0
[1,2,3,4,5]                                             0
[1,3,2]                                                 1
[1,6,2,7,3,8,4,9,5,10]                                  1
[1,3,5,7,2,4,6]                                         2
[1,8,6,4,2,9,7,5,3,10]                                  2
[1,9,8,7,6,5,4,3,2,10]                                  3
[1,5,9,4,8,3,7,2,6,10]                                  4
[1,3,5,7,9,2,4,6,8]                                     5
[1,6,11,5,10,4,9,3,8,2,7]                               6
[1,10,19,9,18,8,17,7,16,6,15,5,14,4,13,3,12,2,11,20]    10
[1,3,5,7,9,11,13,15,17,19,2,4,6,8,10,12,14,16,18,20]    17
[1,141,32,172,63,203,94,234,125,16,156,47,187,78,218,109,249,140,31,171,62,202,93,233,124,15,155,46,186,77,217,108,248,139,30,170,61,201,92,232,123,14,154,45,185,76,216,107,247,138,29,169,60,200,91,231,122,13,153,44,184,75,215,106,246,137,28,168,59,199,90,230,121,12,152,43,183,74,214,105,245,136,27,167,58,198,89,229,120,11,151,42,182,73,213,104,244,135,26,166,57,197,88,228,119,10,150,41,181,72,212,103,243,134,25,165,56,196,87,227,118,9,149,40,180,71,211,102,242,133,24,164,55,195,86,226,117,8,148,39,179,70,210,101,241,132,23,163,54,194,85,225,116,7,147,38,178,69,209,100,240,131,22,162,53,193,84,224,115,6,146,37,177,68,208,99,239,130,21,161,52,192,83,223,114,5,145,36,176,67,207,98,238,129,20,160,51,191,82,222,113,4,144,35,175,66,206,97,237,128,19,159,50,190,81,221,112,3,143,34,174,65,205,96,236,127,18,158,49,189,80,220,111,2,142,33,173,64,204,95,235,126,17,157,48,188,79,219,110,250]
45

• One or two test cases with an odd length and an output greater than 0 would be nice. It's easy to mess the riffle in such cases if you have to write the riffle code by yourself instead of relying on builtins. – Olivier Grégoire Mar 11 at 14:21
• @OlivierGrégoire The [1,3,5,7,9,2,4,6,8] is of length 9, but I will add a few more for lengths 7 and 11 perhaps. EDIT: Added the test cases [1,3,5,7,2,4,6] = 2 (length 7) and [1,6,11,5,10,4,9,3,8,2,7] = 6 (length 11). Hope that helps. – Kevin Cruijssen Mar 11 at 14:27
• My bad: I was sure the test case you mentioned was of size 8. But thanks for the extra test cases. – Olivier Grégoire Mar 11 at 15:37
• Question as currently formulated seems "wrong"... a single riffle shuffle should result in the first and last cards changing, unless you're pulling some kind of con trick! i.e. [6,1,7,2,8,3,9,4,10,5] after a single shuffle of 10 cards. – Steve Mar 12 at 11:05
• @Steve I guess you're kinda right. Riffle-shuffling in general simply interleaves two halves, so both [1,6,2,7,3,8,4,9,5,10] or [6,1,7,2,8,3,9,4,10,5] are possible. In my challenge it does mean that the top card will always remain the top card, so it's indeed a bit of a con-trick.. I've never seen someone irl use only riffle-shuffles to shuffle a deck of cards however. Usually they also use other type of shuffles in between. Anyway, it's too late to change the challenge now, so for the sake of this challenge the top card will always remain the top card after a riffle-shuffle. – Kevin Cruijssen Mar 12 at 11:20

# Jelly, 8 bytes

ŒœẎ$ƬiṢ’  Try it online! ### How? ŒœẎ$ƬiṢ’ - Link: list of integers A
Ƭ    - collect up until results are no longer unique...
$- last two links as a monad: Œœ - odds & evens i.e. [a,b,c,d,...] -> [[a,c,...],[b,d,...]] Ẏ - tighten -> [a,c,...,b,d,...] Ṣ - sort A i - first (1-indexed) index of sorted A in collected shuffles ’ - decrement  # JavaScript (ES6), 44 bytes Shorter version suggested by @nwellnhof Expects a deck with 1-indexed cards as input. f=(a,x=1)=>a[x]-2&&1+f(a,x*2%(a.length-1|1))  Try it online! Given a deck $$\[c_0,\ldots,c_{L-1}]\$$ of length $$\L\$$, we define: $$x_n=\begin{cases} 2^n\bmod L&\text{if }L\text{ is odd}\\ 2^n\bmod (L-1)&\text{if }L\text{ is even}\\ \end{cases}$$ And we look for $$\n\$$ such that $$\c_{x_n}=2\$$. # JavaScript (ES6), 57 52 50 bytes Expects a deck with 0-indexed cards as input. f=(a,x=1,k=a.length-1|1)=>a[1]-x%k&&1+f(a,x*-~k/2)  Try it online! ### How? Since JS is lacking native support for extracting array slices with a custom stepping, simulating the entire riffle-shuffle would probably be rather costly (but to be honest, I didn't even try). However, the solution can also be found by just looking at the 2nd card and the total number of cards in the deck. Given a deck of length $$\L\$$, this code looks for $$\n\$$ such that: $$c_2\equiv\left(\frac{k+1}{2}\right)^n\pmod k$$ where $$\c_2\$$ is the second card and $$\k\$$ is defined as: $$k=\begin{cases} L&\text{if }L\text{ is odd}\\ L-1&\text{if }L\text{ is even}\\ \end{cases}$$ # Python 2, 39 bytes f=lambda x:x[1]-2and-~f(x[::2]+x[1::2])  Try it online! -4 thanks to Jonathan Allan. • Save four bytes with f=lambda x:2!=x[1]and-~f(x[::2]+x[1::2]) – Jonathan Allan Mar 11 at 14:48 • @JonathanAllan Oh, of course! Well... != can be -. ;-) – Erik the Outgolfer Mar 11 at 14:49 • Ah, yeah caveat emptor :D (or just x[1]>2 I guess) – Jonathan Allan Mar 11 at 14:50 # Perl 6, 34 32 bytes -2 bytes thanks to Jo King {(.[(2 X**^$_)X%$_-1+|1]...2)-1}  Try it online! Similar to Arnauld's approach. The index of the second card after n shuffles is 2**n % k with k defined as in Arnauld's answer. # APL (Dyalog Unicode), 352623 22 bytesSBCS {⍵≡⍳≢⍵:0⋄1+∇⍵[⍒2|⍳⍴⍵]}  Try it online! Thanks to Adám for the help, Erik the Outgolfer for -3 and ngn for -1. The TIO link contains two test cases. Explanation: {⍵≡⍳≢⍵:0⋄1+∇⍵[⍒2|⍳⍴⍵]} {⍵≡⍳≢⍵:0⋄1+∇⍵[⍒2|⍳⍴⍵]} ⍝ function takes one argument: ⍵, the array ⍵≡⍳≢⍵ ⍝ if the array is sorted: ⍵≡⍳≢⍵ ⍝ array = 1..length(array) :0 ⍝ then return 0 ⋄ ⍝ otherwise 1+ ⍝ increment ∇ ⍝ the value of the recursive call with this argument: ⍵[ ] ⍝ index into the argument with these indexes: ⍳⍴⍵ ⍝ - generate a range from 1 up to the size of ⍵ 2| ⍝ - %2: generate a binary mask like [1 0 1 0 1 0] ⍒ ⍝ - grade (sorts but returns indexes instead of values), so we have the indexes of all the 1s first, then the 0s.  ¹ • Count the recursion depth for -3. – Erik the Outgolfer Mar 11 at 13:26 • @EriktheOutgolfer Much better, thanks! – Ven Mar 11 at 13:29 • ∧/2≤/⍵ -> ⍵≡⍳≢⍵ – ngn Mar 12 at 15:58 • @ngn didn't realize the array had no holes. Thanks! – Ven Mar 12 at 16:59 # Perl 6, 36 34 32 bytes -2 bytes thanks to nwellnhof $!={.[1]-2&&$!(.sort:{$++%2})+1}


Try it online!

Reverse riffle shuffles by sorting by the index modulo 2 until the list is sorted, then returns the length of the sequence.

It's funny, I don't usually try the recursive approach for Perl 6, but this time it ended up shorter than the original.

### Explanation:

$!={.[1]-2&&$!(.sort:{$++%2})+1}$!={                           }   # Assign the anonymous code block to $! .[1]-2&& # While the list is not sorted$!(             )      # Recursively call the function on
.sort:{$++%2} # It sorted by the parity of each index +1 # And return the number of shuffles  # R, 5855 45 bytes a=scan();while(a[2]>2)a=matrix(a,,2,F<-F+1);F  Try it online! Simulates the sorting process. Input is 1-indexed, returns FALSE for 0. • Very nice! I was working on a similar approach but using a recursive function, which didn't work out as golfy. – user2390246 Mar 12 at 11:55 # 05AB1E (legacy), 9 bytes [DāQ#ι˜]N  Try it online! Explanation [ # ] # loop until ā # the 1-indexed enumeration of the current list D Q # equals a copy of the current list ι˜ # while false, uninterleave the current list and flatten N # push the iteration index N as output  • I didn't even knew it was possible to output the index outside the loop in the legacy. I thought it would be 0 again at that point, just like in the new 05AB1E version. Nice answer! Shorter than my 10-byter using the unshuffle-builtin Å≠ that inspired this challenge. :) – Kevin Cruijssen Mar 11 at 10:31 • @KevinCruijssen: Interesting. I didn't know there was an unshuffle. In this instance it's the same as my version, but unshuffle maintains dimensions on 2D arrays. – Emigna Mar 11 at 11:18 # Java (JDK), 59 bytes a->{int c=0;for(;a[(1<<c)%(a.length-1|1)]>2;)c++;return c;}  Try it online! Works reliably only for arrays with a size less than 31 or solutions with less than 31 iterations. For a more general solution, see the following solution with 63 bytes: a->{int i=1,c=0;for(;a[i]>2;c++)i=i*2%(a.length-1|1);return c;}  Try it online! ## Explanation In a riffle, the next position is the previous one times two modulo either length if it's odd or length - 1 if it's even. So I'm iterating over all indices using this formula until I find the value 2 in the array. ## Credits • 163 bytes by using two times x.clone() instead of A.copyOf(x,l). – Kevin Cruijssen Mar 11 at 12:58 • 64 bytes – Arnauld Mar 11 at 13:57 • @Arnauld Thanks! I had a hard time figuring how to simplify that "length if odd else length - 1" – Olivier Grégoire Mar 11 at 13:59 • @Arnauld Oh! My new algorithm is actually the same as yours... And I spent half an hour figuring it out by myself... – Olivier Grégoire Mar 11 at 14:04 • More precisely, it's equivalent to an improvement over my original algorithm found by @nwellnhof. – Arnauld Mar 11 at 14:10 # J, 28 26 bytes -2 bytes thanks to Jonah!  1#@}.(\:2|#\)^:(2<1{])^:a:  Try it online! Inspired be Ven's APL solution. ## Explanation:  ^: ^:a: while (2<1{]) the 1-st (zero-indexed) element is greater than 2 ( ) do the following and keep the intermediate results i.@# make a list form 0 to len-1 2| find modulo 2 of each element /: sort the argument according the list of 0's and 1's 1 }. drop the first row of the result #@ and take the length (how many rows -> steps)  # K (ngn/k), 25 bytes Thanks to ngn for the advice and for his K interpreter! {#1_{~2=x@1}{x@<2!!#x}\x}  Try it online! • converge-iterate, then drop one, and count - this leads to shorter code – ngn Mar 12 at 16:28 • @ngn. So, similar to my J solution - I'll try it later, thanks! – Galen Ivanov Mar 12 at 18:16 • 1#@}.(\:2|#\)^:(2<1{])^:a: for 26 bytes – Jonah Apr 22 at 3:29 • @Jonah Thank you! – Galen Ivanov Apr 22 at 3:54 # APL(NARS), chars 49, bytes 98 {0{∧/¯1↓⍵≤1⌽⍵:⍺⋄(⍺+1)∇⍵[d],⍵[i∼d←↑¨i⊂⍨2∣i←⍳≢⍵]}⍵}  why use in the deepest loop, one algo that should be nlog(n), when we can use one linear n? just for few bytes more? [⍵≡⍵[⍋⍵] O(nlog n) and the confront each element for see are in order using ∧/¯1↓⍵≤1⌽⍵ O(n)]test:  f←{0{∧/¯1↓⍵≤1⌽⍵:⍺⋄(⍺+1)∇⍵[d],⍵[i∼d←↑¨i⊂⍨2∣i←⍳≢⍵]}⍵} f ,1 0 f 1 2 3 0 f 1,9,8,7,6,5,4,3,2,10 3 f 1,3,5,7,9,11,13,15,17,19,2,4,6,8,10,12,14,16,18,20 17  • That’s the first time I’ve seen someone differentiate between characters and bytes 👍. It always bugs me when I see Unicode characters and they claim that it’s one byte per character. This 😠 is not one byte! – Kerndog73 Mar 11 at 21:31 • @Kerndog73 All is number, but in APL think characters are not numbers... (they seems element in AV array) – RosLuP Mar 11 at 23:41 # Ruby, 42 bytes f=->d,r=1{d[r]<3?0:1+f[d,r*2%(1|~-d.max)]}  Try it online! ### How: Search for number 2 inside the array: if it's in second position, the deck hasn't been shuffled, otherwise check the positions where successive shuffles would put it. # R, 70 72 bytes x=scan();i=0;while(any(x>sort(x))){x=c(x[y<-seq(x)%%2>0],x[!y]);i=i+1};i  Try it online! Now handles the zero shuffle case. • @user2390246 fair point. Adjusted accordingly – Nick Kennedy Mar 12 at 12:03 # C (GCC) 64 63 bytes -1 byte from nwellnhof i,r;f(c,v)int*v;{for(i=r=1;v[i]>2;++r)i=i*2%(c-1|1);return~-r;}  This is a drastically shorter answer based on Arnauld's and Olivier Grégoire's answers. I'll leave my old solution below since it solves the slightly more general problem of decks with cards that are not contiguous. Try it online # C (GCC) 162 bytes a[999],b[999],i,r,o;f(c,v)int*v;{for(r=0;o=1;++r){for(i=c;i--;(i&1?b:a)[i/2]=v[i])o=(v[i]>v[i-1]|!i)&o;if(o)return r;for(i+=o=c+1;i--;)v[i]=i<o/2?a[i]:b[i-o/2];}}  Try it online a[999],b[999],i,r,o; //pre-declare variables f(c,v)int*v;{ //argument list for(r=0;o=1;++r){ //major loop, reset o (ordered) to true at beginning, increment number of shuffles at end for(i=c;i--;(i&1?b:a)[i/2]=v[i]) //loop through v, split into halves a/b as we go o=(v[i]>v[i-1]|!i)&o; //if out of order set o (ordered) to false if(o) //if ordered return r; //return number of shuffles //note that i==-1 at this point for(i+=o=c+1;i--;)//set i=c and o=c+1, loop through v v[i]=i<o/2?a[i]:b[i-o/2];//set first half of v to a, second half to b } }  # R, 85 bytes s=scan();u=sort(s);k=0;while(any(u[seq(s)]!=s)){k=k+1;u=as.vector(t(matrix(u,,2)))};k  Try it online. Explanation Stupid (brute force) method, much less elegant than following the card #2. Instead of unshuffling the input s we start with a sorted vector u that we progressively shuffle until it is identical with s. This gives warnings (but shuffle counts are still correct) for odd lengths of input due to folding an odd-length vector into a 2-column matrix; in that case, in R, missing data point is filled by recycling of the first element of input. The loop will never terminate if we provide a vector that cannot be unshuffled. Addendum: you save one byte if unshuffling instead. Unlike the answer above, there is no need to transpose with t(), however, ordering is byrow=TRUE which is why T appears in matrix(). # R, 84 bytes s=scan();u=sort(s);k=0;while(any(s[seq(u)]!=u)){k=k+1;s=as.vector(matrix(s,,2,T))};k  Try it online! • I took the liberty of fixing your title and adding a TIO-link for the test cases (based on the other R answer), and also verified your answer works as intended, so +1 from me and welcome to PPCG! :) – Kevin Cruijssen Mar 13 at 10:52 # PowerShell, 116 114 108 84 78 bytes -24 bytes thanks to Erik the Outgolfer's solution. -6 bytes thanks to mazzy. param($a)for(;$a[1]-2){$n++;$t=@{};$a|%{$t[$j++%2]+=,$_};$a=$t.0+$t.1;$j=0}+$n


Try it online!

• you can save a bit more: Try it online! – mazzy Mar 18 at 4:49
• @muzzy, you're right again :) thanks – Andrei Odegov Mar 18 at 7:41

# Wolfram Language (Mathematica), 62 bytes

c=0;While[Sort[a]!=a,a=a[[1;;-1;;2]]~Join~a[[2;;-1;;2]];c++];c


Try it online!

# Explanation

The input list is a . It is unriffled and compared with the sorted list until they match.

# Red, 8779 78 bytes

func[b][c: 0 while[b/2 > 2][c: c + 1 b: append extract b 2 extract next b 2]c]


Try it online!

map{push@{$_%2},$_}0..$#F;$_=0;++$_,@s=sort{$a-$b}@F=@F[@0,@1]while"@F"ne"@s"  Try it online! # Japt, 13 11 bytes Taking my shiny, new, very-work-in-progress interpreter for a test drive. g1 Í©ÒßUñÏu  g1 Í©ÒßUñÏu :Implicit input of integer array U g1 :Get the element at 0-based index 1 Í :Subtract from 2 © :Logical AND with Ò : Negation of bitwise NOT of ß : A recursive call to the programme with input Uñ : U sorted Ï : By 0-based indices u : Modulo 2  • This interpreter looks super cool. – recursive Mar 11 at 23:51 # Pyth, 18 bytes L?SIb0hys%L2>Bb1 y  Try it online! -2 thanks to @Erik the Outgolfer. The script has two line: the first one defines a function y, the second line calls y with the implicit Q (evaluated stdin) argument. L?SIb0hys%L2>Bb1 L function y(b) ? if... SIb the Invariant b == sort(b) holds 0 return 0 h otherwise increment... y ...the return of a recursive call with: B the current argument "bifurcated", an array of: b - the original argument > 1 - same with the head popped off L map... % 2 ...take only every 2nd value in each array s and concat them back together  ¹ # PowerShell, 627170 66 bytes +9 bytes when Test cases with an even number of elements added. -1 byte with splatting. -4 bytes: wrap the expression with $i,$j to a new scope. for($a=$args;$a[1]-2;$a=&{($a|?{++$j%2})+($a|?{$i++%2})}){$n++}+\$n


Try it online!

# Python 3, 40 bytes

f=lambda x:x[1]-2and 1+f(x[::2]+x[1::2])  # 1-based
f=lambda x:x[1]-1and 1+f(x[::2]+x[1::2])  # 0-based


Try it online!

I need to refresh the page more frequently: missed Erik the Outgolfer's edit doing a similar trick =)