538 Riddler Express - helping Santa

538_santa.R

library(tidyverse)

# define amount of trials for the Monte Carlo Simulation
trial_n <- 1e5
order <- NULL
n_smaller <- NULL

# generate random order for each trial
for (trial in 1:trial_n) {
  set.seed(181231 + trial)
  
  order_trial <- c(1, sample(2:9, 8))
  
  n_smaller_trial <- NULL
  
  for (i in 1:9) {
    smaller <- sum(order_trial[1:i] < order_trial[i])
    n_smaller_trial <- c(n_smaller_trial, smaller)
  }
  
  order <- c(order, order_trial)
  n_smaller <- c(n_smaller, n_smaller_trial)
}

# create tibble with simulations
simulations <-
  crossing(trial = 1:trial_n,
           position = 1:9) %>%
  mutate(order = order,
         n_smaller = n_smaller,
         time_max = order,
         time = time_max - n_smaller)

simulations_summary <-
  simulations %>%
  group_by(trial) %>%
  summarize(time_total = sum(time))

# answer to Riddler Express question
mean_time_total <- mean(simulations_summary$time_total)

# plot mean time per position
position_order <- c(9:1)

simulations %>%
  group_by(position) %>%
  summarize(time_mean = mean(time)) %>%
  mutate(position = factor(position, levels = position_order)) %>%
  ggplot(aes(x = position, y = time_mean)) +
  geom_col() +
  scale_y_continuous(limits = c(0, 8)) +
  labs(y = "mean time to find right reindeer [min]")