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 word   Pfannkuchen   meaning   pancake.



Related tasks

   Number reversal game
   Sorting algorithms/Pancake sort


 
topswops <- function(x){ 
  i <- 0
  while(x[1] != 1){
    first <- x[1]
    if(first == length(x)){
      x <- rev(x)
    } else{
      x <- c(x[first:1], x[(first+1):length(x)])
    }
    i <- i + 1
  }
  return(i)
}
 
library(iterpc)
 
result <- NULL
 
for(i in 1:10){
  I <- iterpc(i, labels = 1:i, ordered = T)
  A <- getall(I)
  A <- data.frame(A)
  A$flips <- apply(A, 1, topswops)
  result <- rbind(result, c(i, max(A$flips)))
}
 