Martin
May 9, 2016
Shut the Box is a dice game of chance where the player tries to eliminate all numbered tiles by rolling two die and flipping over tiles which add up to the sum of the two die. To “Shut the Box”, all tiles must be flipped over. The player is unable to shut the box if the sum of two die add to a number which cannot be reached by any open tiles in the box.
I want to know how likely it is to win a game of Shut the Box" by closing all numbers. In particular, I’m interested about what happens when the box goes all the way up to 12.
Given that the box has 12 numbers, and that we are able to roll only one die when the sum of the numbers left in the box is less than 6, what is the probability of shutting the box completely?
I wrote a python script that simulates playing a round of Shut the Box, and ran it 1 million times to find the chance of closing the box (scoring 0).
Short answer: You can shut the box ~ 0.41% of the time, or about 1 in 250 tries
It may be possible to improve on the strategy I use, and if anyone wants to do so I’d love to hear about it!
library(ggplot2)theme_set(theme_bw(base_size = 20))#setwd("Projects/shut_the_box")a <- read.csv("box_scores.csv", header = FALSE)p <- ggplot(a, aes(x=V1)) + geom_histogram(color="black", fill="white", binwidth = 1) + xlab("Score") + ylab("Count") + geom_vline(aes(xintercept=mean(V1)), color="red", linetype="dashed", size=1) + annotate("Text", x = 60, y = 33000, label = paste("Mean = ", round(mean(a$V1), 1)), size = 10)
Here is a histogram of 1 million simulated rounds of playing Shut the Box. The average score (the red dashed line) is around 36:
That’s cool. It’s also kind of interesting that there peaks where some scores are more likely than others.
Even though the average score was around 36, the most common score in Shut the Box is a 33. At least in the way that I’m telling the computer to close numbers.
library(dplyr); library(knitr); library(data.table); library(scales)Percent <- sort(summary(as.factor(a$V1)), decreasing = TRUE)df <- as.data.frame(Percent)setDT(df, keep.rownames=TRUE)df$percent <- transmute(df, Percent = percent(Percent / sum(Percent)))x <- percent((df[rn %in% c(1:12),][, sum(Percent)]) / df[,sum(Percent)])df <- df[, .(rn, percent)]colnames(df) <- c("Score", "Percent")
The following table is a breakdown of the most common scores by percent that they showed up in 1 million simulations. I think one of the most interesting things is that shutting the box (scoring 0) is not the least likely outcome. According to the following table, it’s more rare that you’d end up with an end score of 9, for example. Some of that has to do with how the computer makes the decisions. But it’s also pretty common that you would roll a 5-9, and if not they can be used to close other numbers, so it’s rare that you would end up with, say, a 9 being the only number left.
So you might be less likely to end up with certain single digit scores than closing the box, but as a whole you have a 3.84% chance of ending on a single digit score compared to 0.41% chance of shutting the box.
| Score | Percent |
|---|
| 33 | 3.82% |
| 34 | 3.20% |
| 42 | 3.14% |
| 23 | 3.04% |
| 32 | 2.97% |
| 35 | 2.89% |
| 36 | 2.84% |
| 43 | 2.78% |
| 38 | 2.75% |
| 37 | 2.73% |
| 41 | 2.73% |
| 31 | 2.70% |
| 40 | 2.69% |
| 39 | 2.59% |
| 44 | 2.59% |
| 30 | 2.38% |
| 24 | 2.34% |
| 45 | 2.33% |
| 26 | 2.21% |
| 29 | 2.19% |
| 28 | 2.18% |
| 46 | 2.14% |
| 25 | 2.13% |
| 27 | 2.10% |
| 47 | 2.09% |
| 22 | 2.07% |
| 48 | 1.94% |
| 50 | 1.82% |
| 49 | 1.75% |
| 51 | 1.68% |
| 21 | 1.68% |
| 12 | 1.44% |
| 53 | 1.41% |
| 52 | 1.40% |
| 54 | 1.24% |
| 20 | 1.24% |
| 55 | 1.04% |
| 19 | 1.02% |
| 13 | 0.96% |
| 57 | 0.94% |
| 17 | 0.91% |
| 56 | 0.91% |
| 15 | 0.91% |
| 18 | 0.90% |
| 14 | 0.85% |
| 16 | 0.85% |
| 59 | 0.80% |
| 58 | 0.71% |
| 11 | 0.67% |
| 64 | 0.53% |
| 61 | 0.48% |
| 60 | 0.46% |
| 62 | 0.42% |
| 0 | 0.41% |
| 10 | 0.39% |
| 9 | 0.26% |
| 63 | 0.23% |
| 65 | 0.23% |
| 66 | 0.19% |
| 8 | 0.18% |
| 1 | 0.16% |
| 68 | 0.15% |
| 7 | 0.14% |
| 6 | 0.14% |
| 70 | 0.12% |
| 2 | 0.12% |
| 3 | 0.12% |
| 5 | 0.11% |
| 4 | 0.11% |
| 71 | 0.09% |
| 76 | 0.08% |
| 67 | 0.07% |
| 72 | 0.04% |
| 69 | 0.04% |
| 73 | 0.03% |
FAQs
The player with the lowest score wins. If you manage to “shut the box”, you win instantly. HISTORY The history of this game is not much more than vague folklore -- No one really knows when and where the game originated. It may have originated in Normandy, France around the 12th Century.
What is the strategy for Shut the Box? ›
The best strategy is to use the combined score for a high tile (7,8,9), if possible, otherwise choose the lowest tile. The success rate for this strategy is 7.9855%. Digital – A player's score at the end of the turn is the number obtained by reading the up digits as a decimal number from left to right.
What are the odds of shutting the box 12? ›
There are no standard rules for the 12 box version, but the analysis given here assumes that you always use 2 dice. As shown below, it is much more difficult to shut all 12 boxes. The probability of winning is only 0.3622 percent. (One chance in 276.)
What is the scorecard in Shut the Box? ›
A player's score is the sum of the numbers remaining uncovered at the end of their turn. The player with the lowest score wins. Another fun way to keep score is to sum up the numbers on the tiles that you have covered.
What is the snake eyes in Shut the Box? ›
In the worst case scenario, a player could roll "snake eyes" (double ones, or sum of two) twice in a row. Since the two would have been closed on the first roll, on the second roll of 2, the game would end, leaving the player with a score of 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 43.
How does Shut the Box end? ›
If anyone succeeds in shutting the box i.e. closing all the numbers, that player wins outright immediately and receives double the stake from all players. Otherwise, after each player has taken one turn, the winner of the round is the player with the lowest score.
What happens if you get a double in Shut the Box? ›
For example, if a 4 and 5 are rolled, they would turn over the 4 and 5 cards. If a double is thrown, the player's turn ends. They can roll the dice again until they can't turn over any more cards. The cards that are left showing are then added and that is their score.
What age is Shut the Box for? ›
A fun mathematics game from Schylling - Ages 4+
Can you play Shut the Box by yourself? ›
The player with the lowest score wins! Shut the Box is usually played with 2-4 players, but you can also play solo. Just compete against your lowest score.
How to play 4 way shut the box? ›
Players can challenge each other for the lowest score. Flip down all the tiles and you've Shut-the-Box! With 4-Way Play, everyone rolls their dice at the same time, flipping down the matching tiles. When no one can flip any more numbers, the person with the lowest score (least not flipped) wins the game!
The object of the game of Shut the Box is to “shut” as many numbers as possible and become the player with the lowest score.
What is the game with two dice and numbers 1 9? ›
An old pub gambling favourite, Shut the Box uses two dice and a special wooden playing tray. The tray features the numbers 1 - 9 in a row, each of which has a hinged or sliding cover. A turn involves repeatedly throwing the dice and shutting or covering a number or pair of numbers every throw.
What's in the box game objective? ›
The aim of the game is to guess what is inside the box- be it real things you've found round the house or using the picture cards included.
How do you beat the box game? ›
In order for the entire group to win the game, they must enter their code into the electronic combination lock before the countdown is over. There are four intriguing cases in each aluminium box which are protected by strong combination locks. These contain codes and mysteries that teams must solve to progress.
What's in the box game speech therapy? ›
Use it for a game of “What's in the Box?” Gather random objects from around the house and place them inside when your child isn't looking. Describe one object at a time. Give clues about the shape, size, color, what it's used for, etc.
Is Shut the Box fun to play? ›
Shut the Box is a fun and easy game to play for all family members. A perfect game to play with your kids or grandchildren.
