Perfect shuffle

From Rosetta Code
Task
Perfect shuffle
You are encouraged to solve this task according to the task description, using any language you may know.

A perfect shuffle (or faro/weave shuffle) means splitting a deck of cards into equal halves, and perfectly interleaving them - so that you end up with the first card from the left half, followed by the first card from the right half, and so on:

7♠ 8♠ 9♠ J♠ Q♠ K♠
7♠  8♠  9♠
  J♠  Q♠  K♠
7♠ J♠ 8♠ Q♠ 9♠ K♠

When you repeatedly perform perfect shuffles on an even-sized deck of unique cards, it will at some point arrive back at its original order. How many shuffles this takes, depends solely on the number of cards in the deck - for example for a deck of eight cards it takes three shuffles:

original:

1 2 3 4 5 6 7 8

after 1st shuffle:

1 5 2 6 3 7 4 8

after 2nd shuffle:

1 3 5 7 2 4 6 8

after 3rd shuffle:

1 2 3 4 5 6 7 8

The Task

  1. Write a function that can perform a perfect shuffle on an even-sized list of values.
  2. Call this function repeatedly to count how many shuffles are needed to get a deck back to its original order, for each of the deck sizes listed under "Test Cases" below.
    • You can use a list of numbers (or anything else that's convenient) to represent a deck; just make sure that all "cards" are unique within each deck.
    • Print out the resulting shuffle counts, to demonstrate that your program passes the test-cases.

Test Cases

input (deck size) output (number of shuffles required)
8 3
24 11
52 8
100 30
1020 1018
1024 10
10000 300

ALGOL 68[edit]

# returns an array of the specified length, initialised to an ascending sequence of integers #
OP DECK = ( INT length )[]INT:
BEGIN
[ 1 : length ]INT result;
FOR i TO UPB result DO result[ i ] := i OD;
result
END # DECK # ;
 
# in-place shuffles the deck as per the task requirements #
# LWB deck is assumed to be 1 #
PROC shuffle = ( REF[]INT deck )VOID:
BEGIN
[ 1 : UPB deck ]INT result;
INT left pos := 1;
INT right pos := ( UPB deck OVER 2 ) + 1;
FOR i FROM 2 BY 2 TO UPB result DO
result[ left pos ] := deck[ i - 1 ];
result[ right pos ] := deck[ i ];
left pos +:= 1;
right pos +:= 1
OD;
FOR i TO UPB deck DO deck[ i ] := result[ i ] OD
END # SHUFFLE # ;
 
# compares two integer arrays for equality #
OP = = ( []INT a, b )BOOL:
IF LWB a /= LWB b OR UPB a /= UPB b
THEN # the arrays have different bounds #
FALSE
ELSE
BOOL result := TRUE;
FOR i FROM LWB a TO UPB a WHILE result := a[ i ] = b[ i ] DO SKIP OD;
result
FI # = # ;
 
# compares two integer arrays for inequality #
OP /= = ( []INT a, b )BOOL: NOT ( a = b );
 
# returns the number of shuffles required to return a deck of the specified length #
# back to its original state #
PROC count shuffles = ( INT length )INT:
BEGIN
[] INT original deck = DECK length;
[ 1 : length ]INT shuffled deck := original deck;
INT count := 1;
WHILE shuffle( shuffled deck );
shuffled deck /= original deck
DO
count +:= 1
OD;
count
END # count shuffles # ;
 
# test the shuffling #
[]INT lengths = ( 8, 24, 52, 100, 1020, 1024, 10 000 );
FOR l FROM LWB lengths TO UPB lengths DO
print( ( whole( lengths[ l ], -8 ) + ": " + whole( count shuffles( lengths[ l ] ), -6 ), newline ) )
OD
Output:
       8:      3
      24:     11
      52:      8
     100:     30
    1020:   1018
    1024:     10
   10000:    300

AutoHotkey[edit]

Shuffle(cards){
n := cards.MaxIndex()/2, res := []
loop % n
res.push(cards[A_Index]), res.push(cards[round(A_Index + n)])
return res
}
Examples:
test := [8, 24, 52, 100, 1020, 1024, 10000]
for each, val in test
{
cards := [], original:=rep:=""
loop, % val
cards.push(A_Index), original .= (original?", ":"") A_Index
while (res <> original)
{
res := ""
for k, v in (cards := Shuffle(cards))
res .= (res?", ":"") v
rep := A_Index
}
result .= val "`t" rep "`n"
}
MsgBox % result
return
Outputs:
8	3
24	11
52	8
100	30
1020	1018
1024	10
10000	300

C[edit]

/* ===> INCLUDES <============================================================*/
#include <stdlib.h>
#include <stdio.h>
#include <string.h>
 
/* ===> CONSTANTS <===========================================================*/
#define N_DECKS 7
const int kDecks[N_DECKS] = { 8, 24, 52, 100, 1020, 1024, 10000 };
 
/* ===> FUNCTION PROTOTYPES <=================================================*/
int CreateDeck( int **deck, int nCards );
void InitDeck( int *deck, int nCards );
int DuplicateDeck( int **dest, const int *orig, int nCards );
int InitedDeck( int *deck, int nCards );
int ShuffleDeck( int *deck, int nCards );
void FreeDeck( int **deck );
 
/* ===> FUNCTION DEFINITIONS <================================================*/
 
int main() {
int i, nCards, nShuffles;
int *deck = NULL;
 
for( i=0; i<N_DECKS; ++i ) {
nCards = kDecks[i];
 
if( !CreateDeck(&deck,nCards) ) {
fprintf( stderr, "Error: malloc() failed!\n" );
return 1;
}
 
InitDeck( deck, nCards );
nShuffles = 0;
 
do {
ShuffleDeck( deck, nCards );
++nShuffles;
} while( !InitedDeck(deck,nCards) );
 
printf( "Cards count: %d, shuffles required: %d.\n", nCards, nShuffles );
 
FreeDeck( &deck );
}
 
return 0;
}
 
int CreateDeck( int **deck, int nCards ) {
int *tmp = NULL;
 
if( deck != NULL )
tmp = malloc( nCards*sizeof(*tmp) );
 
return tmp!=NULL ? (*deck=tmp)!=NULL : 0; /* (?success) (:failure) */
}
 
void InitDeck( int *deck, int nCards ) {
if( deck != NULL ) {
int i;
 
for( i=0; i<nCards; ++i )
deck[i] = i;
}
}
 
int DuplicateDeck( int **dest, const int *orig, int nCards ) {
if( orig != NULL && CreateDeck(dest,nCards) ) {
memcpy( *dest, orig, nCards*sizeof(*orig) );
return 1; /* success */
}
else {
return 0; /* failure */
}
}
 
int InitedDeck( int *deck, int nCards ) {
int i;
 
for( i=0; i<nCards; ++i )
if( deck[i] != i )
return 0; /* not inited */
 
return 1; /* inited */
}
 
int ShuffleDeck( int *deck, int nCards ) {
int *copy = NULL;
 
if( DuplicateDeck(&copy,deck,nCards) ) {
int i, j;
 
for( i=j=0; i<nCards/2; ++i, j+=2 ) {
deck[j] = copy[i];
deck[j+1] = copy[i+nCards/2];
}
 
FreeDeck( &copy );
return 1; /* success */
}
else {
return 0; /* failure */
}
}
 
void FreeDeck( int **deck ) {
if( *deck != NULL ) {
free( *deck );
*deck = NULL;
}
}
 
Output:
Cards count: 8, shuffles required: 3.
Cards count: 24, shuffles required: 11.
Cards count: 52, shuffles required: 8.
Cards count: 100, shuffles required: 30.
Cards count: 1020, shuffles required: 1018.
Cards count: 1024, shuffles required: 10.
Cards count: 10000, shuffles required: 300.


Press "Enter" to quit...

C++[edit]

 
#include <iostream>
#include <algorithm>
#include <vector>
 
int pShuffle( int t ) {
std::vector<int> v, o, r;
 
for( int x = 0; x < t; x++ ) {
o.push_back( x + 1 );
}
 
r = o;
int t2 = t / 2 - 1, c = 1;
 
while( true ) {
v = r;
r.clear();
 
for( int x = t2; x > -1; x-- ) {
r.push_back( v[x + t2 + 1] );
r.push_back( v[x] );
}
 
std::reverse( r.begin(), r.end() );
 
if( std::equal( o.begin(), o.end(), r.begin() ) ) return c;
c++;
}
}
 
int main() {
int s[] = { 8, 24, 52, 100, 1020, 1024, 10000 };
for( int x = 0; x < 7; x++ ) {
std::cout << "Cards count: " << s[x] << ", shuffles required: ";
std::cout << pShuffle( s[x] ) << ".\n";
}
return 0;
}
 
Output:
Cards count: 8, shuffles required: 3.
Cards count: 24, shuffles required: 11.
Cards count: 52, shuffles required: 8.
Cards count: 100, shuffles required: 30.
Cards count: 1020, shuffles required: 1018.
Cards count: 1024, shuffles required: 10.
Cards count: 10000, shuffles required: 300.

Clojure[edit]

(defn perfect-shuffle [deck]
(let [half (split-at (/ (count deck) 2) deck)]
(interleave (first half) (last half))))
 
(defn solve [deck-size]
(let [original (range deck-size)
trials (drop 1 (iterate perfect-shuffle original))
predicate #(= original %)]
(println (format "%5s: %s" deck-size
(inc (some identity (map-indexed (fn [i x] (when (predicate x) i)) trials)))))))
 
(map solve [8 24 52 100 1020 1024 10000])
Output:
    8: 3
   24: 11
   52: 8
  100: 30
 1020: 1018
 1024: 10
10000: 300

Common Lisp[edit]

(defun perfect-shuffle (deck)
(let* ((half (floor (length deck) 2))
(left (subseq deck 0 half))
(right (nthcdr half deck)))
(mapcan #'list left right)))
 
(defun solve (deck-size)
(loop with original = (loop for n from 1 to deck-size collect n)
for trials from 1
for deck = original then shuffled
for shuffled = (perfect-shuffle deck)
until (equal shuffled original)
finally (format t "~5D: ~4D~%" deck-size trials)))
 
(solve 8)
(solve 24)
(solve 52)
(solve 100)
(solve 1020)
(solve 1024)
(solve 10000)
Output:
    8:    3
   24:   11
   52:    8
  100:   30
 1020: 1018
 1024:   10
10000:  300

EchoLisp[edit]

 
;; shuffler : a permutation vector which interleaves both halves of deck
(define (make-shuffler n)
(let ((s (make-vector n)))
(for ((i (in-range 0 n 2))) (vector-set! s i (/ i 2)))
(for ((i (in-range 0 n 2))) (vector-set! s (1+ i) (+ (/ n 2) (vector-ref s i))))
s))
 
;; output : (n . # of shuffles needed to go back)
(define (magic-shuffle n)
(when (odd? n) (error "magic-shuffle:odd input" n))
(let [(deck (list->vector (iota n))) ;; (0 1 ... n-1)
(dock (list->vector (iota n))) ;; keep trace or init deck
(shuffler (make-shuffler n))]
 
(cons n (1+
(for/sum ((i Infinity)) ; (in-naturals missing in EchoLisp v2.9)
(vector-permute! deck shuffler) ;; permutes in place
#:break (eqv? deck dock) ;; compare to first
1)))))
 
Output:
 
map magic-shuffle '(8 24 52 100 1020 1024 10000))
((8 . 3) (24 . 11) (52 . 8) (100 . 30) (1020 . 1018) (1024 . 10) (10000 . 300))
 
;; Let's look in the On-line Encyclopedia of Integer Sequences
;; Given a list of numbers, the (oeis ...) function looks for a sequence
 
(lib 'web)
Lib: web.lib loaded.
map magic-shuffle (range 2 18 2))
((2 . 1) (4 . 2) (6 . 4) (8 . 3) (10 . 6) (12 . 10) (14 . 12) (16 . 4))
(oeis '(1 2 4 3 6 10 12 4))
→ Sequence A002326 found
 

Elixir[edit]

Translation of: Ruby
defmodule Perfect do
def shuffle(n) do
start = Enum.to_list(1..n)
m = div(n, 2)
shuffle(start, magic_shuffle(start, m), m, 1)
end
 
defp shuffle(start, start, _, step), do: step
defp shuffle(start, deck, m, step) do
shuffle(start, magic_shuffle(deck, m), m, step+1)
end
 
defp magic_shuffle(deck, len) do
{left, right} = Enum.split(deck, len)
Enum.zip(left, right)
|> Enum.map(&Tuple.to_list/1)
|> List.flatten
end
end
 
Enum.each([8, 24, 52, 100, 1020, 1024, 10000], fn n ->
step = Perfect.shuffle(n)
IO.puts "#{n} : #{step}"
end)
Output:
8 : 3
24 : 11
52 : 8
100 : 30
1020 : 1018
1024 : 10
10000 : 300

Go[edit]

package main
 
import "fmt"
 
type Deck struct {
Cards []int
length int
}
 
func NewDeck(deckSize int) (res *Deck){
if deckSize % 2 != 0{
panic("Deck size must be even")
}
res = new(Deck)
res.Cards = make([]int, deckSize)
res.length = deckSize
for i,_ := range res.Cards{
res.Cards[i] = i
}
return
}
func (d *Deck)shuffleDeck(){
tmp := make([]int,d.length)
for i := 0;i <d.length/2;i++ {
tmp[i*2] = d.Cards[i]
tmp[i*2+1] = d.Cards[d.length / 2 + i]
}
d.Cards = tmp
}
func (d *Deck) isEqualTo(c Deck) (res bool) {
if d.length != c.length {
panic("Decks aren't equally sized")
}
res = true
for i, v := range d.Cards{
if v != c.Cards[i] {
res = false
}
}
return
}
 
 
func main(){
for _,v := range []int{8,24,52,100,1020,1024,10000} {
fmt.Printf("Cards count: %d, shuffles required: %d\n",v,ShufflesRequired(v))
}
}
 
func ShufflesRequired(deckSize int)(res int){
deck := NewDeck(deckSize)
Ref := *deck
deck.shuffleDeck()
res++
for ;!deck.isEqualTo(Ref);deck.shuffleDeck(){
res++
}
return
}
Output:
Cards count: 8, shuffles required: 3
Cards count: 24, shuffles required: 11
Cards count: 52, shuffles required: 8
Cards count: 100, shuffles required: 30
Cards count: 1020, shuffles required: 1018
Cards count: 1024, shuffles required: 10
Cards count: 10000, shuffles required: 300 

Haskell[edit]

shuffle :: [a] -> [a]   
shuffle lst = let (a,b) = splitAt (length lst `div` 2) lst
in foldMap (\(x,y) -> [x,y]) $ zip a b
 
findCycle :: Eq a => (a -> a) -> a -> [a]
findCycle f x = takeWhile (/= x) $ iterate f (f x)
 
main = mapM_ report [ 8, 24, 52, 100, 1020, 1024, 10000 ]
where
report n = putStrLn ("deck of " ++ show n ++ " cards: "
++ show (countSuffles n) ++ " shuffles!")
countSuffles n = 1 + length (findCycle shuffle [1..n])
Output:
deck of 8 cards: 3 shuffles!
deck of 24 cards: 11 shuffles!
deck of 52 cards: 8 shuffles!
deck of 100 cards: 30 shuffles!
deck of 1020 cards: 1018 shuffles!
deck of 1024 cards: 10 shuffles!
deck of 10000 cards: 300 shuffles!

J[edit]

The shuffle routine:

   shuf=: /: $ /:@$ 0 1"_

Here, the phrase ($ $ 0 1"_) would generate a sequence of 0s and 1s the same length as the argument sequence:

   ($ $ 0 1"_) 'abcdef'
0 1 0 1 0 1

And we can use grade up (/:) to find the indices which would sort the argument sequence so that the values in the positions corresponding to our generated zeros would come before the values in the positions corresponding to our ones.

   /: ($ $ 0 1"_) 'abcdef'
0 2 4 1 3 5

But we can use grade up again to find what would have been the original permutation (grade up is a self inverting function for this domain).

   /:/: ($ $ 0 1"_) 'abcdef'
0 3 1 4 2 5

And, that means it can also sort the original sequence into that order:

   shuf 'abcdef'
adbecf
shuf 'abcdefgh'
aebfcgdh

And this will work for sequences of arbitrary length.

(The rest of the implementation of shuf is pure syntactic sugar - you can use J's dissect and trace facilities to see the details if you are trying to learn the language.)

Meanwhile, the cycle length routine could look like this:

   shuflen=:  [: *./ #@>@C.@[email protected]

Here, we first generate a list of integers of the required length in their natural order. We then reorder them using our shuf function, find the cycles which result, find the lengths of each of these cycles then find the least common multiple of those lengths.

So here is the task example (with most of the middle trimmed out to avoid crashing the rosettacode wiki implementation):

   shuflen"0 }.2*i.5000
1 2 4 3 6 10 12 4 8 18 6 11 20 18 28 5 10 12 36 12 20 14 12 23 21 8 52 20 18 ... 4278 816 222 1332 384

Task example:

  ('deck size';'required shuffles'),(; shuflen)&> 8 24 52 100 1020 1024 10000
┌─────────┬─────────────────┐
│deck size│required shuffles│
├─────────┼─────────────────┤
83
├─────────┼─────────────────┤
2411
├─────────┼─────────────────┤
528
├─────────┼─────────────────┤
10030
├─────────┼─────────────────┤
10201018
├─────────┼─────────────────┤
102410
├─────────┼─────────────────┤
10000300
└─────────┴─────────────────┘

Note that the implementation of shuf defines a behavior for odd length "decks". Experimentation shows that cycle length for an odd length deck is often the same as the cycle length for an even length deck which is one "card" longer.

Java[edit]

Works with: Java version 8
import java.util.Arrays;
import java.util.stream.IntStream;
 
public class PerfectShuffle {
 
public static void main(String[] args) {
int[] sizes = {8, 24, 52, 100, 1020, 1024, 10_000};
for (int size : sizes)
System.out.printf("%5d : %5d%n", size, perfectShuffle(size));
}
 
static int perfectShuffle(int size) {
if (size % 2 != 0)
throw new IllegalArgumentException("size must be even");
 
int half = size / 2;
int[] a = IntStream.range(0, size).toArray();
int[] original = a.clone();
int[] aa = new int[size];
 
for (int count = 1; true; count++) {
System.arraycopy(a, 0, aa, 0, size);
 
for (int i = 0; i < half; i++) {
a[2 * i] = aa[i];
a[2 * i + 1] = aa[i + half];
}
 
if (Arrays.equals(a, original))
return count;
}
}
}
    8 :     3
   24 :    11
   52 :     8
  100 :    30
 1020 :  1018
 1024 :    10
10000 :   300

JavaScript[edit]

ES6[edit]

(() => {
'use strict';
 
// shuffleCycleLength :: Int -> Int
const shuffleCycleLength = deckSize =>
firstCycle(shuffle, range(1, deckSize))
.all.length;
 
// shuffle :: [a] -> [a]
const shuffle = xs =>
concat(zip.apply(null, splitAt(div(length(xs), 2), xs)));
 
// firstycle :: Eq a => (a -> a) -> a -> [a]
const firstCycle = (f, x) =>
until(
m => EqArray(x, m.current),
m => {
const fx = f(m.current);
return {
current: fx,
all: m.all.concat([fx])
};
}, {
current: f(x),
all: [x]
}
);
 
// Two arrays equal ?
// EqArray :: [a] -> [b] -> Bool
const EqArray = (xs, ys) => {
const [nx, ny] = [xs.length, ys.length];
return nx === ny ? (
nx > 0 ? (
xs[0] === ys[0] && EqArray(xs.slice(1), ys.slice(1))
) : true
) : false;
};
 
// GENERIC FUNCTIONS
 
// zip :: [a] -> [b] -> [(a,b)]
const zip = (xs, ys) =>
xs.slice(0, Math.min(xs.length, ys.length))
.map((x, i) => [x, ys[i]]);
 
// concat :: [[a]] -> [a]
const concat = xs => [].concat.apply([], xs);
 
// splitAt :: Int -> [a] -> ([a],[a])
const splitAt = (n, xs) => [xs.slice(0, n), xs.slice(n)];
 
// div :: Num -> Num -> Int
const div = (x, y) => Math.floor(x / y);
 
// until :: (a -> Bool) -> (a -> a) -> a -> a
const until = (p, f, x) => {
const go = x => p(x) ? x : go(f(x));
return go(x);
}
 
// range :: Int -> Int -> [Int]
const range = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);
 
// length :: [a] -> Int
// length :: Text -> Int
const length = xs => xs.length;
 
// maximumBy :: (a -> a -> Ordering) -> [a] -> a
const maximumBy = (f, xs) =>
xs.reduce((a, x) => a === undefined ? x : (
f(x, a) > 0 ? x : a
), undefined);
 
// transpose :: [[a]] -> [[a]]
const transpose = xs =>
xs[0].map((_, iCol) => xs.map((row) => row[iCol]));
 
// show :: a -> String
const show = x => JSON.stringify(x, null, 2);
 
// replicateS :: Int -> String -> String
const replicateS = (n, s) => {
let v = s,
o = '';
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};
 
// justifyRight :: Int -> Char -> Text -> Text
const justifyRight = (n, cFiller, strText) =>
n > strText.length ? (
(replicateS(n, cFiller) + strText)
.slice(-n)
) : strText;
 
// TEST
return transpose(transpose([
['Deck', 'Shuffles']
].concat(
[8, 24, 52, 100, 1020, 1024, 10000]
.map(n => [n.toString(), shuffleCycleLength(n)
.toString()
])))
.map(col => { // Right-justified number columns
const width = length(
maximumBy((a, b) => length(a) - length(b), col)
) + 2;
 
return col.map(x => justifyRight(width, ' ', x));
}))
.map(row => row.join(''))
.join('\n');
})();
Output:
   Deck  Shuffles
      8         3
     24        11
     52         8
    100        30
   1020      1018
   1024        10
  10000       300

Kotlin[edit]

// version 1.1.1
 
fun areSame(a: IntArray, b: IntArray): Boolean {
for (i in 0 until a.size) if (a[i] != b[i]) return false
return true
}
 
fun perfectShuffle(a: IntArray): IntArray {
var b = IntArray(a.size)
val hSize = a.size / 2
for (i in 0 until hSize) b[i * 2] = a[i]
var j = 1
for (i in hSize until a.size) {
b[j] = a[i]
j += 2
}
return b
}
 
fun countShuffles(a: IntArray): Int {
require(a.size >= 2 && a.size % 2 == 0)
var b = a
var count = 0
while (true) {
val c = perfectShuffle(b)
count++
if (areSame(a, c)) return count
b = c
}
}
 
fun main(args: Array<String>) {
println("Deck size Num shuffles")
println("--------- ------------")
val sizes = intArrayOf(8, 24, 52, 100, 1020, 1024, 10000)
for (size in sizes) {
val a = IntArray(size) { it }
val count = countShuffles(a)
println("${"%-9d".format(size)} $count")
}
}
Output:
Deck size  Num shuffles
---------  ------------
8             3
24            11
52            8
100           30
1020          1018
1024          10
10000         300

Lua[edit]

-- Perform weave shuffle
function shuffle (cards)
local pile1, pile2 = {}, {}
for card = 1, #cards / 2 do table.insert(pile1, cards[card]) end
for card = (#cards / 2) + 1, #cards do table.insert(pile2, cards[card]) end
cards = {}
for card = 1, #pile1 do
table.insert(cards, pile1[card])
table.insert(cards, pile2[card])
end
return cards
end
 
-- Return boolean indicating whether or not the cards are in order
function inOrder (cards)
for k, v in pairs(cards) do
if k ~= v then return false end
end
return true
end
 
-- Count the number of shuffles needed before the cards are in order again
function countShuffles (deckSize)
local deck, count = {}, 0
for i = 1, deckSize do deck[i] = i end
repeat
deck = shuffle(deck)
count = count + 1
until inOrder(deck)
return count
end
 
-- Main procedure
local testCases = {8, 24, 52, 100, 1020, 1024, 10000}
print("Input", "Output")
for _, case in pairs(testCases) do print(case, countShuffles(case)) end
Output:
Input   Output
8       3
24      11
52      8
100     30
1020    1018
1024    10
10000   300


MATLAB[edit]

Function:

   function [New]=PerfectShuffle(Nitems, Nturns)
   if mod(Nitems,2)==0 %only if even number
       X=1:Nitems; %define deck
       for c=1:Nturns %defines one shuffle
           X=reshape(X,Nitems/2,2)'; %split the deck in two and stack halves
           X=X(:)'; %mix the halves
       end
       New=X; %result of multiple shufflings
   end
   end


Main:

   Result=[]; %vector to store results 
   Q=[8, 24, 52, 100, 1020, 1024, 10000]; %queries
   for n=Q %for each query
       Same=0; %initialize comparison
       T=0; %initialize number of shuffles
       while ~Same %while the result is not the original query
           T=T+1; %one more shuffle
           R=PerfectShuffle(n,T); %result of shuffling the query
           Same=~(any(R-(1:n))); %same vector as the query
       end %when getting the same vector
       Result=[Result;T]; %collect results
   end
   disp([Q', Result])

Output:

          8           3
         24          11
         52           8
        100          30
       1020        1018
       1024          10
      10000         300

Oforth[edit]

: shuffle(l)     l size 2 / dup l left swap l right zip expand ;
: nbShuffles(l) 1 l while( shuffle dup l <> ) [ 1 under+ ] drop ;
Output:
>[ 8, 24, 52, 100, 1020, 1024, 10000 ] map(#[ seq nbShuffles ]) .
[3, 11, 8, 30, 1018, 10, 300] ok

PARI/GP[edit]

This example is in need of improvement:
The task description was updated; please update this solution accordingly and then remove this template.
magic(v)=vector(#v,i,v[if(i%2,1,#v/2)+i\2]);
shuffles_slow(n)=my(v=[1..n],o=v,s=1);while((v=magic(v))!=o,s++);s;
shuffles(n)=znorder(Mod(2,n-1));
vector(5000,n,shuffles_slow(2*n))
Output:
%1 = [1, 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, 20, 18, 28, 5, 10, 12, 36, 12,
 20, 14, 12, 23, 21, 8, 52, 20, 18, 58, 60, 6, 12, 66, 22, 35, 9, 20, 30, 39, 54
, 82, 8, 28, 11, 12, 10, 36, 48, 30, 100, 51, 12, 106, 36, 36, 28, 44, 12, 24, 1
10, 20, 100, 7, 14, 130, 18, 36, 68, 138, 46, 60, 28, 42, 148, 15, 24, 20, 52, 5
2, 33, 162, 20, 83, 156, 18, 172, 60, 58, 178, 180, 60, 36, 40, 18, 95, 96, 12,
196, 99, 66, 84, 20, 66, 90, 210, 70, 28, 15, 18, 24, 37, 60, 226, 76, 30, 29, 9
2, 78, 119, 24, 162, 84, 36, 82, 50, 110, 8, 16, 36, 84, 131, 52, 22, 268, 135,
12, 20, 92, 30, 70, 94, 36, 60, 136, 48, 292, 116, 90, 132, 42, 100, 60, 102, 10
2, 155, 156, 12, 316, 140, 106, 72, 60, 36, 69, 30, 36, 132, 21, 28, 10, 147, 44
, 346, 348, 36, 88, 140, 24, 179, 342, 110, 36, 183, 60, 156, 372, 100, 84, 378,
 14, 191, 60, 42, 388, 88, 130, 156, 44, 18, 200, 60, 108, 180, 204, 68, 174, 16
4, 138, 418, 420, 138, 40, 60, 60, 43, 72, 28, 198, 73, 42, 442, 44, 148, 224, 2
0, 30, 12, 76, 72, 460, 231, 20, 466, 66, 52, 70, 180, 156, 239, 36, 66, 48, 243
, 162, 490, 56, 60, 105, 166, 166, 251, 100, 156, 508, 9, 18, 204, 230, 172, 260
, 522, 60, 40, 253, 174, 60, 212, 178, 210, 540, 180, 36, 546, 60, 252, 39, 36,
556, 84, 40, 562, 28, 54, 284, 114, 190, 220, 144, 96, 246, 260, 12, 586, 90, 19
6, 148, 24, 198, 299, 25, 66, 220, 303, 84, 276, 612, 20, 154, 618, 198, 33, 500
, 90, 72, 45, 210, 28, 84, 210, 64, 214, 28, 323, 290, 30, 652, 260, 18, 658, 66
0, 24, 36, 308, 74, 60, 48, 180, 676, 48, 226, 22, 68, 76, 156, 230, 30, 276, 40
, 58, 700, 36, 92, 300, 708, 78, 55, 60, 238, 359, 51, 24, 140, 121, 486, 56, 24
4, 84, 330, 246, 36, 371, 148, 246, 318, 375, 50, 60, 756, 110, 380, 36, 24, 348
, 384, 16, 772, 20, 36, 180, 70, 252, 52, 786, 262, 84, 60, 52, 796, 184, 66, 90
, 132, 268, 404, 270, 270, 324, 126, 12, 820, 411, 20, 826, 828, 92, 168, 332, 9
0, 419, 812, 70, 156, 330, 94, 396, 852, 36, 428, 858, 60, 431, 172, 136, 390, 1
32, 48, 300, 876, 292, 55, 882, 116, 443, 21, 270, 414, 356, 132, 140, 104,[+++]

(By default gp won't show more than 25 lines of output, though an arbitrary amount can be printed or written to a file; use print, write, or default(lines, 100) to show more.)

Perl[edit]

use List::Util qw(all);
 
sub perfect_shuffle {
my $mid = @_ / 2;
map { @_[$_, $_ + $mid] } 0..($mid - 1);
}
 
for my $size (8, 24, 52, 100, 1020, 1024, 10000) {
 
my @shuffled = my @deck = 1 .. $size;
my $n = 0;
do { $n++; @shuffled = perfect_shuffle(@shuffled) }
until all { $shuffled[$_] == $deck[$_] } 0..$#shuffled;
 
printf "%5d cards: %4d\n", $size, $n;
}
Output:
    8 cards:    3
   24 cards:   11
   52 cards:    8
  100 cards:   30
 1020 cards: 1018
 1024 cards:   10
10000 cards:  300

Perl 6[edit]

Translation of: Perl
sub perfect-shuffle (@deck) {
my $mid = @deck / 2;
flat @deck[0 ..^ $mid] Z @deck[$mid .. *];
}
 
for 8, 24, 52, 100, 1020, 1024, 10000 -> $size {
my @deck = ^$size;
my $n;
repeat until [<] @deck {
$n++;
@deck = perfect-shuffle @deck;
}
 
printf "%5d cards: %4d\n", $size, $n;
}
Output:
    8 cards:    3
   24 cards:   11
   52 cards:    8
  100 cards:   30
 1020 cards: 1018
 1024 cards:   10
10000 cards:  300

Phix[edit]

function perfect_shuffle(sequence deck)
integer mp = length(deck)/2
sequence res = deck
integer k = 1
for i=1 to mp do
res[k] = deck[i] k += 1
res[k] = deck[i+mp] k += 1
end for
return res
end function
 
constant testsizes = {8, 24, 52, 100, 1020, 1024, 10000}
for i=1 to length(testsizes) do
sequence deck = tagset(testsizes[i])
sequence work = perfect_shuffle(deck)
integer count = 1
while work!=deck do
work = perfect_shuffle(work)
count += 1
end while
printf(1,"%5d cards: %4d\n", {testsizes[i],count})
end for
Output:
    8 cards:    3
   24 cards:   11
   52 cards:    8
  100 cards:   30
 1020 cards: 1018
 1024 cards:   10
10000 cards:  300

PicoLisp[edit]

(de perfectShuffle (Lst)
(mapcan '((B A) (list A B))
(cdr (nth Lst (/ (length Lst) 2)))
Lst ) )
 
(for N (8 24 52 100 1020 1024 10000)
(let (Lst (range 1 N) L Lst Cnt 1)
(until (= Lst (setq L (perfectShuffle L)))
(inc 'Cnt) )
(tab (5 6) N Cnt) ) )

Output:

    8     3
   24    11
   52     8
  100    30
 1020  1018
 1024    10
10000   300

Python[edit]

 
import doctest
import random
 
 
def flatten(lst):
"""
>>> flatten([[3,2],[1,2]])
[3, 2, 1, 2]
"""

return [i for sublst in lst for i in sublst]
 
def magic_shuffle(deck):
"""
>>> magic_shuffle([1,2,3,4])
[1, 3, 2, 4]
"""

half = len(deck) // 2
return flatten(zip(deck[:half], deck[half:]))
 
def after_how_many_is_equal(shuffle_type,start,end):
"""
>>> after_how_many_is_equal(magic_shuffle,[1,2,3,4],[1,2,3,4])
2
"""

 
start = shuffle_type(start)
counter = 1
while start != end:
start = shuffle_type(start)
counter += 1
return counter
 
def main():
doctest.testmod()
 
print("Length of the deck of cards | Perfect shuffles needed to obtain the same deck back")
for length in (8, 24, 52, 100, 1020, 1024, 10000):
deck = list(range(length))
shuffles_needed = after_how_many_is_equal(magic_shuffle,deck,deck)
print("{} | {}".format(length,shuffles_needed))
 
 
if __name__ == "__main__":
main()
 
 

Reversed shuffle or just calculate how many shuffles are needed:

def mul_ord2(n):
# directly calculate how many shuffles are needed to restore
# initial order: 2^o mod(n-1) == 1
if n == 2: return 1
 
n,t,o = n-1,2,1
while t != 1:
t,o = (t*2)%n,o+1
return o
 
def shuffles(n):
a,c = list(range(n)), 0
b = a
 
while True:
# Reverse shuffle; a[i] can be taken as the current
# position of the card with value i. This is faster.
a = a[0:n:2] + a[1:n:2]
c += 1
if b == a: break
return c
 
for n in range(2, 10000, 2):
#print(n, mul_ord2(n))
print(n, shuffles(n))

Racket[edit]

#lang racket/base
(require racket/list)
 
(define (perfect-shuffle l)
(define-values (as bs) (split-at l (/ (length l) 2)))
(foldr (λ (a b d) (list* a b d)) null as bs))
 
(define (perfect-shuffles-needed n)
(define-values (_ rv)
(for/fold ((d (perfect-shuffle (range n))) (i 1))
((_ (in-naturals))
#:break (apply < d))
(values (perfect-shuffle d) (add1 i))))
rv)
 
(module+ test
(require rackunit)
(check-equal? (perfect-shuffle '(1 2 3 4)) '(1 3 2 4))
 
(define (test-perfect-shuffles-needed n e)
(define psn (perfect-shuffles-needed n))
(printf "Deck size:\t~a\tShuffles needed:\t~a\t(~a)~%" n psn e)
(check-equal? psn e))
 
(for-each test-perfect-shuffles-needed
'(8 24 52 100 1020 1024 10000)
'(3 11 8 30 1018 10 300)))
Output:
Deck size:	8	Shuffles needed:	3	(3)
Deck size:	24	Shuffles needed:	11	(11)
Deck size:	52	Shuffles needed:	8	(8)
Deck size:	100	Shuffles needed:	30	(30)
Deck size:	1020	Shuffles needed:	1018	(1018)
Deck size:	1024	Shuffles needed:	10	(10)
Deck size:	10000	Shuffles needed:	300	(300)

REXX[edit]

unoptimized[edit]

/*REXX program performs a  "perfect shuffle"  for a number of  even numbered  decks.    */
parse arg X /*optional list of test cases from C.L.*/
if X='' then X=8 24 52 100 1020 1024 10000 /*Not specified? Then use the default.*/
w=length(word(X, words(X))) /*used for right─aligning the numbers. */
 
do j=1 for words(X); y=word(X,j) /*use numbers in the test suite (list).*/
 
do k=1 for y; @.k=k; end /*k*/ /*generate a deck to be used (shuffled)*/
do t=1 until eq(); call magic; end /*t*/ /*shuffle until before equals after.*/
 
say 'deck size:' right(y,w)"," right(t,w) 'perfect shuffles.'
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
eq: do ?=1 for y; if @.?\==? then return 0; end; return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/
magic: z=0 /*set the Z pointer (used as index).*/
h=y%2 /*get the half─way (midpoint) pointer. */
do s=1 for h; z=z+1; h=h+1 /*traipse through the card deck pips. */
 !.z=@.s; z=z+1 /*assign left half; then bump pointer. */
 !.z=@.h /* " right " */
end /*s*/ /*perform a perfect shuffle of the deck*/
 
do r=1 for y; @.r=!.r; end /*re─assign to the original card deck. */
return

output   (abbreviated)   when using the default input:

deck size:     8,     3 perfect shuffles.
deck size:    24,    11 perfect shuffles.
deck size:    52,     8 perfect shuffles.
deck size:   100,    30 perfect shuffles.
deck size:  1020,  1018 perfect shuffles.
deck size:  1024,    10 perfect shuffles.
deck size: 10000,   300 perfect shuffles.

optimized[edit]

This REXX version takes advantage that the 1st and last cards of the deck don't change.

/*REXX program does a  "perfect shuffle"  for a number of  even  numbered  decks.       */
parse arg X /*optional list of test cases from C.L.*/
if X='' then X=8 24 52 100 1020 1024 10000 /*Not specified? Use default.*/
w=length(word(X, words(X))) /*used for right─aligning the numbers. */
 
do j=1 for words(X); y=word(X,j) /*use numbers in the test suite (list).*/
 
do k=1 for y; @.k=k; end /*generate a deck to be shuffled (used)*/
do t=1 until eq(); call magic; end /*shuffle until before equals after.*/
 
say 'deck size:' right(y,w)"," right(t,w) 'perfect shuffles.'
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
eq: do ?=1 for y; if @.?\==? then return 0; end; return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/
magic: z=1; h=y%2 /*H is (half─way) pointer.*/
do L=3 by 2 for h-1; z=z+1; !.L=@.z; end /*assign left half of deck.*/
do R=2 by 2 for h-1; h=h+1; !.R=@.h; end /* " right " " " */
do a=2 for y-2; @.a=!.a; end /*re─assign──►original deck*/
return

output   is the same as the 1st version.

Ruby[edit]

def perfect_shuffle(deck_size = 52)
deck = (0...deck_size).to_a
shuffled_deck = [deck.first(deck_size / 2), deck.last(deck_size / 2)]
1.step do |i|
return i if deck == (shuffled_deck = shuffled_deck.transpose.flatten)
shuffled_deck = [shuffled_deck.shift(deck_size / 2), shuffled_deck]
end
end
 
[8, 24, 52, 100, 1020, 1024, 10000].each do |i| puts "Perfect Shuffles Required for Deck Size #{i}: #{perfect_shuffle(i)}" end
Output:
Perfect Shuffles Required for Deck Size 8: 3
Perfect Shuffles Required for Deck Size 24: 11
Perfect Shuffles Required for Deck Size 52: 8
Perfect Shuffles Required for Deck Size 100: 30
Perfect Shuffles Required for Deck Size 1020: 1018
Perfect Shuffles Required for Deck Size 1024: 10
Perfect Shuffles Required for Deck Size 10000: 300

Sidef[edit]

Translation of: Perl
func perfect_shuffle(deck) {
var a = deck/2
a[0] ~Z a[1] -> flatten
}
 
[8, 24, 52, 100, 1020, 1024, 10000].each { |size|
 
var deck = @(1..size);
var shuffled = deck;
 
var n = 0;
loop {
++n;
shuffled = perfect_shuffle(shuffled);
 
shuffled.each_index { |i|
shuffled[i] == deck[i] || goto :NEXT;
}
 
break;
@:NEXT;
}
 
printf("%5d cards: %4d\n", size, n);
}
Output:
    8 cards:    3
   24 cards:   11
   52 cards:    8
  100 cards:   30
 1020 cards: 1018
 1024 cards:   10
10000 cards:  300

Tcl[edit]

Using tcltest to include an executable test case ..

namespace eval shuffle {
 
proc perfect {deck} {
if {[llength $deck]%2} {
return -code error "Deck must be of even length!"
}
set split [expr {[llength $deck]/2}]
set top [lrange $deck 0 $split-1]
set btm [lrange $deck $split end]
foreach a $top b $btm {
lappend res $a $b
}
return $res
}
 
proc cycle_length {transform deck} {
set d $deck
while 1 {
set d [$transform $d]
incr i
if {$d eq $deck} {return $i}
}
return $i
}
 
proc range {a {b ""}} {
if {$b eq ""} {
set b $a; set a 0
}
set res {}
while {$a < $b} {
lappend res $a
incr a
}
return $res
}
 
}
 
set ::argv {}
package require tcltest
tcltest::test "Test perfect shuffle cycles" {} -body {
lmap size {8 24 52 100 1020 1024 10000} {
shuffle::cycle_length perfect [range $size]
}
} -result {3 11 8 30 1018 10 300}

zkl[edit]

fcn perfectShuffle(numCards){
deck,shuffle,n,N:=numCards.pump(List),deck,0,numCards/2;
do{ shuffle=shuffle[0,N].zip(shuffle[N,*]).flatten(); n+=1 }
while(deck!=shuffle);
n
}
foreach n in (T(8,24,52,100,1020,1024,10000)){
println("%5d : %d".fmt(n,perfectShuffle(n)));
}
Output:
    8 : 3
   24 : 11
   52 : 8
  100 : 30
 1020 : 1018
 1024 : 10
10000 : 300