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Topswops is a card game created by John Conway in the 1970's.

Assume you have a particular permutation of a set of n cards numbered 1..n on both of their faces, for example the arrangement of four cards given by [2, 4, 1, 3] where the leftmost card is on top. A round is composed of reversing the first m cards where m is the value of the topmost card. rounds are repeated until the topmost card is the number 1 and the number of swaps is recorded. For our example the swaps produce:

    [2, 4, 1, 3]    # Initial shuffle
    [4, 2, 1, 3]
    [3, 1, 2, 4]
    [2, 1, 3, 4]
    [1, 2, 3, 4]

For a total of four swaps from the initial ordering to produce the terminating case where 1 is on top.

For a particular number n of cards, topswops(n) is the maximum swaps needed for any starting permutation of the n cards.

Task

The task is to generate and show here a table of n vs topswops(n) for n in the range 1..10 inclusive.

Note

Topswops is also known as Fannkuch from the German Pfannkuchen meaning pancake.

Cf.

C

An algorithm that doesn't go through all permutations, per Knuth tAoCP 7.2.1.2 exercise 107 (possible bad implementation on my part notwithstanding): <lang c>#include <stdio.h>

include <string.h>

typedef struct { char v[16]; } deck; typedef unsigned int uint;

uint n, d, best[16];

void tryswaps(deck *a, uint f, uint s) {

define A a->v
define B b.v

if (d > best[n]) best[n] = d; while (1) { if ((A[s] == s || (A[s] == -1 && !(f & 1U << s))) && (d + best[s] >= best[n] || A[s] == -1)) break;

if (d + best[s] <= best[n]) return; if (!--s) return; }

d++; deck b = *a; for (uint i = 1, k = 2; i <= s; k <<= 1, i++) { if (A[i] != i && (A[i] != -1 || (f & k))) continue;

for (uint j = B[0] = i; j--;) B[i - j] = A[j]; tryswaps(&b, f | k, s); } d--; }

int main(void) { deck x; memset(&x, -1, sizeof(x)); x.v[0] = 0;

for (n = 1; n < 13; n++) { tryswaps(&x, 1, n - 1); printf("%2d: %d\n", n, best[n]); }

return 0; }</lang>

D

Permutations generator from: http://rosettacode.org/wiki/Permutations#Faster_Lazy_Version <lang d>import std.stdio, std.algorithm, std.range, permutations2;

uint topswops(in uint n) in {

   assert (n > 0 && n < 32);

} body {

   static uint flip(uint[] deck) pure nothrow {
       uint[32] temp = void;
       temp[0 .. deck.length] = deck[];
       uint count = 0;
       for (auto t0 = temp[0]; t0; t0 = temp[0]) {
           temp[0 .. t0 + 1].reverse(); // Slow with DMD.
           count++;
       }
       return count;
   }

   return iota(n).array().permutations!0().map!flip().reduce!max();

}

void main() {

   foreach (i; 1 .. 11)
       writefln("%2d: %d", i, topswops(i));

}</lang>

Output:

D: Faster Version

Translation of: C

<lang d>import std.stdio, std.typetuple;

template Range(int start, int stop) {

   static if (stop <= start)
       alias TypeTuple!() Range;
   else
       alias TypeTuple!(Range!(start, stop - 1), stop - 1) Range;

}

__gshared uint[32] best;

uint topswops(size_t n)() nothrow {

   static assert(n > 0 && n < best.length);
   size_t d = 0;

   alias T = byte;
   alias Deck = T[n];

   void trySwaps(in ref Deck deck, in uint f) nothrow {
       if (d > best[n])
           best[n] = d;

       foreach_reverse (immutable i; Range!(0, n)) {

if ((deck[i] == i || (deck[i] == -1 && !(f & 1U<<i))) && (d + best[i] >= best[n] || deck[i] == -1)) break;

           if (d + best[i] <= best[n])
               return;
       }

       Deck deck2 = void;
       foreach (immutable i; Range!(0, n)) // Copy.
           deck2[i] = deck[i];

       d++;
       foreach (immutable i; Range!(1, n)) {
           enum uint k = 1U << i;
           if (deck2[i] == -1) {
               if (f & k)
                   continue;
           } else if (deck2[i] != i)
               continue;

           deck2[0] = cast(T)i;
           foreach_reverse (immutable j; Range!(0, i))
               deck2[i - j] = deck[j]; // Reverse copy.
           trySwaps(deck2, f | k);
       }
       d--;
   }

   best[n] = 0;
   Deck deck0 = -1;
   deck0[0] = 0;
   trySwaps(deck0, 1);
   return best[n];

}

void main() {

   foreach (i; Range!(1, 14))
       writefln("%2d: %d", i, topswops!i());

}</lang>

Output:

This version also uses templates to speed up the computation.

Go

<lang go>// Adapted from http://www-cs-faculty.stanford.edu/~uno/programs/topswops.w // at Donald Knuth's web site. Algorithm credited there to Pepperdine // and referenced to Mathematical Gazette 73 (1989), 131-133. package main

import "fmt"

const ( // array sizes

   maxn = 10 // max number of cards
   maxl = 50 // upper bound for number of steps

)

func main() {

   for i := 1; i <= maxn; i++ {
       fmt.Printf("%d: %d\n", i, steps(i))
   }

}

func steps(n int) int {

   var a, b [maxl][maxn + 1]int
   var x [maxl]int
   a[0][0] = 1
   var m int
   for l := 0; ; {
       x[l]++
       k := int(x[l])
       if k >= n {
           if l <= 0 {
               break
           }
           l--
           continue
       }
       if a[l][k] == 0 {
           if b[l][k+1] != 0 {
               continue
           }
       } else if a[l][k] != k+1 {
           continue
       }
       a[l+1] = a[l]
       for j := 1; j <= k; j++ {
           a[l+1][j] = a[l][k-j]
       }
       b[l+1] = b[l]
       a[l+1][0] = k + 1
       b[l+1][k+1] = 1
       if l > m-1 {
           m = l + 1
       }
       l++
       x[l] = 0
   }
   return m

}</lang>

Output:

Haskell

<lang Haskell>import Data.List (permutations)

topswops :: Int -> Int topswops n = foldl max 0 $ map topswops' $ permutations [1..n]

topswops' :: [Int] -> Int topswops' xa@(1:_) = 0 topswops' xa@(x:_) = 1 + topswops' reordered

   where
       reordered = reverse (take x xa) ++ drop x xa

main = mapM_

       (\x -> putStrLn $ show x ++ ":\t" ++ show (topswops x))
       [1..10]</lang>

Output:

J

Solution:<lang j> swops =: ((|.@:{. , }.)~ {.)^:a:</lang> Example (from task introduction):<lang j> swops 2 4 1 3 2 4 1 3 4 2 1 3 3 1 2 4 2 1 3 4 1 2 3 4</lang> Example (topswops of all permutations of the integers 1..10):<lang j> (,. _1 + ! >./@:(#@swops@A. >:)&i. ])&> 1+i.10

10 38</lang>

Alternative

Interpretation of faster C version.

<lang j>trySwop=: 4 :0 'n d f s'=. y best=: n (d>.{)`[`]} best whilst. s=.s-1 do.

 if. s (= +. _1 = ]) s{x do. break. end.
 if. (d+s{best) <: n{best do. return. end.

end. d=.d+1 t=._1=tx=.s{. }. x for_i. 1+I.0= ((-.t)*.tx~:1+i.s) +. t *. 0~:f 17 b. 2^1+i.s do.

((|.i{.x) (1+i.i) } i 0 } x) trySwop n,d,s,~f 23 b. 2^i

end. d=.d-1 )

topSwops=: 3 :0 best=: 0$~y+1 for_n. i=.1+i.y do. (0, _1$~y) trySwop n, 0, 1, n-1 end. i,.}. best )</lang>

Output <lang j> topSwops 10

10 38</lang>

Java

Translation of: D

<lang java>public class Topswops {

   static final int maxBest = 32;
   static int[] best;

   static private void trySwaps(int[] deck, int f, int d, int n) {
       if (d > best[n])
           best[n] = d;

       for (int i = n - 1; i >= 0; i--) {
           if (deck[i] == -1 || deck[i] == i)
               break;
           if (d + best[i] <= best[n])
               return;
       }

       int[] deck2 = deck.clone();
       for (int i = 1; i < n; i++) {
           final int k = 1 << i;
           if (deck2[i] == -1) {
               if ((f & k) != 0)
                   continue;
           } else if (deck2[i] != i)
               continue;

           deck2[0] = i;
           for (int j = i - 1; j >= 0; j--)
               deck2[i - j] = deck[j]; // Reverse copy.
           trySwaps(deck2, f | k, d + 1, n);
       }
   }

   static int topswops(int n) {
       assert(n > 0 && n < maxBest);
       best[n] = 0;
       int[] deck0 = new int[n + 1];
       for (int i = 1; i < n; i++)
           deck0[i] = -1;
       trySwaps(deck0, 1, 0, n);
       return best[n];
   }

   public static void main(String[] args) {
       best = new int[maxBest];
       for (int i = 1; i < 11; i++)
           System.out.println(i + ": " + topswops(i));
   }

}</lang>

Output:

Perl 6

<lang perl6>sub postfix:<!>(@a) {

   @a == 1
       ?? [@a]
       !! do for @a -> $a {
               [ $a, @$_ ] for @a.grep(* != $a)!
          }

}

sub swops(@a is copy) {

   my $count = 0;
   until @a[0] == 1 {
       @a[ ^@a[0] ] .= reverse;
       $count++;
   }
   return $count;

} sub topswops($n) { [max] map &swops, (1 .. $n)! }

say "$_ {topswops $_}" for 1 .. 10;</lang>

Output follows that of Python.

Python

This solution uses cards numbered from 0..n-1 and variable p0 is introduced as a speed optimisation <lang python>>>> from itertools import permutations >>> def f1(p): i, p0 = 0, p[0] while p0: i += 1 p0 += 1 p[:p0] = p[:p0][::-1] p0 = p[0] return i

>>> def fannkuch(n): return max(f1(list(p)) for p in permutations(range(n)))

>>> for n in range(1, 11): print(n,fannkuch(n))

1 0 2 1 3 2 4 4 5 7 6 10 7 16 8 22 9 30 10 38 >>> </lang>

Python: Faster Version

Translation of: C

<lang python>try:

   import psyco
   psyco.full()

except ImportError:

   pass

best = [0] * 16

def try_swaps(deck, f, s, d, n):

   if d > best[n]:
       best[n] = d

   i = 0
   k = 1 << s
   while s:
       k >>= 1
       s -= 1
       if deck[s] == -1 or deck[s] == s:
           break
       i |= k
       if (i & f) == i and d + best[s] <= best[n]:
           return d
   s += 1

   deck2 = list(deck)
   k = 1
   for i2 in xrange(1, s):
       k <<= 1
       if deck2[i2] == -1:
           if f & k: continue
       elif deck2[i2] != i2:
           continue

       deck[i2] = i2
       deck2[:i2 + 1] = reversed(deck[:i2 + 1])
       try_swaps(deck2, f | k, s, 1 + d, n)

def topswops(n):

   best[n] = 0
   deck0 = [-1] * 16
   deck0[0] = 0
   try_swaps(deck0, 1, n, 0, n)
   return best[n]

for i in xrange(1, 13):

   print "%2d: %d" % (i, topswops(i))</lang>

Output:

XPL0

<lang XPL0>code ChOut=8, CrLf=9, IntOut=11; int N, Max, Card1(16), Card2(16);

proc Topswop(D); \Conway's card swopping game int D; \depth of recursion int I, J, C, T; [if D # N then \generate N! permutations of 1..N in Card1

    [for I:= 0 to N-1 do
       [for J:= 0 to D-1 do    \check if object (letter) already used
           if Card1(J) = I+1 then J:=100;
       if J < 100 then
           [Card1(D):= I+1;    \card number not used so append it
           Topswop(D+1);       \recurse next level deeper
           ];
       ];
    ]

else [\determine number of topswops to get card 1 at beginning

    for I:= 0 to N-1 do Card2(I):= Card1(I);   \make working copy of deck
       C:= 0;                  \initialize swop counter
       while Card2(0) # 1 do
           [I:= 0;  J:= Card2(0)-1;
           while I < J do
               [T:= Card2(I);  Card2(I):= Card2(J);  Card2(J):= T;
               I:= I+1;  J:= J-1;
               ];
           C:= C+1;
           ];  
    if C>Max then Max:= C;
    ];

];

[for N:= 1 to 10 do

   [Max:= 0;
   Topswop(0);
   IntOut(0, N);  ChOut(0, ^ );  IntOut(0, Max);  CrLf(0);
   ];

]</lang>

Output:

XPL0: Faster Version

Translation of: C

<lang XPL0>code CrLf=9, IntOut=11, Text=12; int N, D, Best(16);

proc TrySwaps(A, F, S); int A, F, S; int B(16), I, J, K; [if D > Best(N) then Best(N):= D; loop [if A(S)=-1 ! A(S)=S then quit;

       if D+Best(S) <= Best(N) then return;
       if S = 0 then quit;
       S:= S-1;
       ];

D:= D+1; for I:= 0 to S do B(I):= A(I); K:= 1; for I:= 1 to S do

       [K:= K<<1;
       if B(I)=-1 & (F&K)=0 ! B(I)=I then
               [J:= I;  B(0):= J;
               while J do [J:= J-1;  B(I-J):= A(J)];
               TrySwaps(B, F!K, S);
               ];
       ];

D:= D-1; ];

int I, X(16); [for I:= 0 to 16-1 do

       [X(I):= -1;  Best(I):= 0];

X(0):= 0; for N:= 1 to 13 do

       [D:= 0;
       TrySwaps(X, 1, N-1);
       IntOut(0, N);  Text(0, ": ");  IntOut(0, Best(N));  CrLf(0);
       ];

]</lang>

Output: