Sometimes I see a Riddler Classic that makes me just want to go attack it with Monte Carlo analysis.

Let’s make a die and roll it 6 times, recording the outcome:

```
die <- 1:6
faces <- sample(seq_along(die), length(die), replace=T) # referring to the length of the die vector will help us if we do the extra credit
die <- die[faces]
die
## [1] 3 2 1 5 1 3
```

That’s 1 iteration. Loops will be necessary since I need the 2nd die to produce the 3rd die, so R’s clever vectorisation won’t help.

I’d like to track a ‘score’ - how far we are from the target of all faces equal. Let’s say the score is the number of distinct numbers still on the die, and the target is a score of 1.

```
update_die <- function(die){
faces <- sample(seq_along(die), length(die), replace=T) # referring to the length of the die vector will help us if we do the extra credit
die[faces]
}
update_score <- function(die){
unique(die) %>%
length()
}
die <- 1:6
score <- update_score(die)
while(last(score) >1){
die <- update_die(die)
score <- c(score, update_score(die))
}
```

Anything we can do 1ce, we can do many times.

```
simulate_1_game <- function(index, size_of_die){
die <- seq_len(size_of_die)
score <- update_score(die)
while(last(score) >1){
die <- update_die(die)
score <- c(score, update_score(die))
}
games <- seq_along(score)
tibble(index=index, score=score, game=games)
}
monte_carlo_d6 <- map_dfr(seq_len(1e4), ~simulate_1_game(.x,6))
```

```
monte_carlo_d6 %>%
ggplot(aes(x=game, after_stat(density))) + geom_freqpoly(binwidth=1) + facet_wrap(~score, ncol=1, scales = "free_y") +
theme + scale_y_continuous(labels=scales::percent)
```

```
monte_carlo_d6 %>%
tabyl(game, score) %>%
pivot_longer(-game, names_to = "score") %>%
ggplot(aes(x=game,y=value,fill=score)) + geom_area() + theme + ggthemes::scale_fill_few()
```

```
monte_carlo_d6 %>%
group_by(index) %>%
summarise(games_until_win = max(game)) %>%
ggplot(aes(x=games_until_win, after_stat(density))) + geom_freqpoly(binwidth = 1) +
theme + scale_y_continuous(labels=scales::percent)
```

```
monte_carlo_d6 %>%
group_by(index) %>%
summarise(games_until_win = max(game)) %>%
tabyl(games_until_win) %>%
mutate(percent = scales::percent(percent)) %>%
knitr::kable()
```

games_until_win | n | percent |
---|---|---|

3 | 124 | 1.2% |

4 | 499 | 5.0% |

5 | 826 | 8.3% |

6 | 979 | 9.8% |

7 | 1018 | 10.2% |

8 | 956 | 9.6% |

9 | 823 | 8.2% |

10 | 757 | 7.6% |

11 | 660 | 6.6% |

12 | 555 | 5.6% |

13 | 439 | 4.4% |

14 | 407 | 4.1% |

15 | 330 | 3.3% |

16 | 297 | 3.0% |

17 | 229 | 2.3% |

18 | 179 | 1.8% |

19 | 137 | 1.4% |

20 | 125 | 1.2% |

21 | 100 | 1.0% |

22 | 85 | 0.8% |

23 | 77 | 0.8% |

24 | 70 | 0.7% |

25 | 56 | 0.6% |

26 | 43 | 0.4% |

27 | 41 | 0.4% |

28 | 38 | 0.4% |

29 | 23 | 0.2% |

30 | 23 | 0.2% |

31 | 17 | 0.2% |

32 | 7 | 0.1% |

33 | 13 | 0.1% |

34 | 15 | 0.2% |

35 | 15 | 0.2% |

36 | 8 | 0.1% |

37 | 2 | 0.0% |

38 | 4 | 0.0% |

39 | 3 | 0.0% |

40 | 4 | 0.0% |

41 | 2 | 0.0% |

42 | 3 | 0.0% |

43 | 2 | 0.0% |

44 | 2 | 0.0% |

45 | 1 | 0.0% |

46 | 2 | 0.0% |

47 | 2 | 0.0% |

48 | 1 | 0.0% |

50 | 1 | 0.0% |

# Other dice

We’d like to keep track of the size of our dice, tweak the simulate function. Just going to track average games until win to save on memory.

```
rm(monte_carlo_d6)
estimate_victory <- function(size_of_die){
original_die <- seq_len(size_of_die)
die <- original_die
dice_pool <- sample(original_die, 1000*size_of_die, replace=T)
play_1_game <- function(...){
n=0L
score <- 10
while(last(score) > 1){
if(length(dice_pool) < size_of_die){
dice_pool <<- sample(original_die, 100*size_of_die, replace=T)
}
n <- n+1L
die <- die[dice_pool[original_die]]
dice_pool <<- dice_pool[-original_die] #Pop 1:n from the dice pool.
score <- c(score, update_score(die))
}
return(n)
}
map_int(1:1000, play_1_game) %>%
mean()
}
results <- future_map_dbl(1:100, estimate_victory)
```

Looks pretty much linear, and might end up more linear if I gave it more simulation time.

```
tibble(size_of_die = 1:100, result=results) %>%
ggplot(aes(x=size_of_die,y=result)) + theme + geom_line()
```