If the probability of flipping is too low, there’s a high risk that nobody flips a coin, but if the probability is too high it approaches summarized %>%Īrrange(desc(pct_free)) # A tibble: 100 x 2 We can thus summarize and visualize the chance of freedom within each probability. In the first ten simulations above, the strategy appeared to work twice (the 6th and 9th observations).Įach value of probability is one strategy: the probability each prisoner decides to use to see if they’ll flip their coin. Notice that the prisoners are set free only if they flip at least one coin and get no tails ( num_flips != 0 & num_tails = 0). The above performs 10 million simulations (100,000 trials for each probability) but because it’s vectorized it’s pretty fast: about 1.5 seconds on my machine. # trial probability num_flips num_tails set_free Set_free = num_flips != 0 & num_tails = 0) Mutate(num_flips = rbinom(n(), 4, probability), Since this probability will the be the same across all four prisoners, we can simulate this for many possible strategies between 1% and 100% using rbinom(). If we all decided to flip the coin, our chance of freedom would be If we all decided not to flip a coin, we’d never get free. This means we’ll all have to take the same strategy, trusting in the fact that all the other logicians will choose the same one. The four prisoners can’t communicate and they’re in symmetrical situations. Simulating four prisonersīefore we jump into our simulation, we can start with a bit of logic. I’ve also posted a 30-minute screencast of how I first approached the simulation and visualization. In an appendix, I’ll show how to get a closed form solution for

I’ll solve this with tidy simulation in R, in particular using one of my favorite functions, tidyr’s crossing(). The only tools you and your fellow prisoners have to aid you are random number generators, which will give each prisoner a random number, uniformly and independently chosen between zero and one. If all flipped coins come up heads, you will all be set free! But if any of the flipped coins comes up tails, or if no one chooses to flip a coin, you will all be doomed to spend the rest of your lives in the castle’s dungeon. But it just so happens that all of you are logicians (of course)….Įach prisoner will be given a fair coin, which can either be fairly flipped one time or returned to the guards without being flipped. You are locked in the dungeon of a faraway castle with three fellow prisoners (i.e., there are four prisoners in total), each in a separate cell with no means of communication. I love 538’s Riddler column, and I’ve enjoyed solving the May 1st puzzle. The “largest stock profit or loss” puzzle.The “knight on an infinite chessboard” puzzle.Fixed the camp setup scene objects so that they don't shift slightly when you mouse over them. Various alternative controls added to different parts of the game. Heavily downsized the file size of the game from 2gb to 300mb. These additions should make the game much more enjoyable to play with either just mouse or just keyboard or a combination of the two. Lastly, you can also play the lockpicking mini-game with either space or by clicking. You can also choose on screen choices by using the number keys. You can now control the animation progression by using space or the arrow keys. I've also added some more controls so that it's much easier to play the game one-handed. The previous build was over 2gb but with this update, I managed to get the game down to around 300mb through some compression and resizing. Not much new content but I did heavily downsize the file size of the game so that it should be much easier to download and store. Here's a quick optimization and quality of life update for you all!

Dungeon tail 0.02