Murder game: what are the chances of a full cycle?

kill_game_cycles.R

## n people draw lots (of each other). What are the chances of a full cycle?

## Draw lots
draw_lots <- function(n, allow_self = F) {
  draw <- sample(n, n)
  while(allow_self == F & any(draw == 1:n)) {
    draw <- sample(n, n)
  }
  return(draw)
}

## 1 begins killing and only stops when they get their own card again.
## By that point, what is their murder count?
## Function will return TRUE if the cycle has length n, FALSE otherwise.
full_cycle <- function(draw) {
  murderer <- 1
  victim <- 0
  murder_count <- 0
  while(victim != 1) {
    victim <- draw[murderer]
    murderer <- victim
    murder_count <- murder_count + 1
  }
  return(murder_count == length(draw))
}

## Run Monte Carlo Simulation
run_mc <- function(n, iterations = 10000, allow_self = F) {
  cycles <- sapply(rep(n, iterations), function(x){full_cycle(draw_lots(x, allow_self = allow_self))})
  chance <- sum(cycles) / length(cycles)
  return(chance)
}

run_mc(18, allow_self = F)

## Probability functions
full_cycle_prob1 = function(n){
  if(n<3){return(1)}
  x = 0
  for(i in 0:(n-2)){
    x = x + (-1)^i * (n-i-1) * factorial(n-2)/factorial(i)
  }
  return(factorial(n-1)/((n-1)*x))
}
full_cycle_prob2 <- function(n) {
  factorial(n-1)/round(factorial(n)/exp(1),0)
}