Diplograph
By Paul Knight

Mind Link

My real life isn’t nearly as sad as I make it sound.

Because I am a huge nerd, I listen to a bunch of “actual play” roleplaying podcasts. They’re recordings of people playing tabletop roleplaying games (think Dungeons and Dragons, but there are a lot of systems out there). It’s almost like I get to share in the fun of having friends.

Anyway, one of the podcasts I listened to for a bit was Party of Four. I don’t really recommend it as the comedians fall back on sexual or gross humor a little too easily, but I did listen to a few episodes.

I’m not sure if Party of Four is using a super stripped down d10 system or something more custom-designed for the podcast, but for the most part there’s very little in the way of mechanics. The occasional skill check is simply the DM and player rolling opposing d10s, with the winner coming out on top. But the one fun mechanic Party of Four uses is the DM will place characters in a super dangerous, unwinnable situation once per session. The only way out is for the player to open the envelope they were given at the start of the session and see what it says…

The envelopes contain challenges ranging from calling 311 and getting the Toronto city services hotline to help the players (in one episode Kayla’s sentient butter stick character is directed to the Dairy Farmers of Ontario) to causing a miracle with an improvised song. One of the challenges is called Mind Link, and I’ll let DM David Dineen-Porter explain:

The Puzzle

In Mind Link, one player is given a sheet of questions and the other player is given a sheet of corresponding answers. Both of them listen to music in their earphones so that they cannot hear one another speak. When one player asks a question, the other player must answer the question without knowing what it is. If the question and answer match they have successfully completed a Mind Link. Four Mind Links are necessary to survive.

And an example of the game:

Kyle
What does Jack Nicholson say to Tom Cruise in A Few Good Men?
Kayla
Molly, you in danger girl. [Bzzt]
Kyle
What did Whoopi Goldberg say to Demi Moore in Ghost?
Kayla
You want the truth? You can't handle the truth! [Bzzt]
Kyle
How did Mr. Rodgers’ song start?
Kayla
It’s a beautiful day in the neighborhood. [Ding!]

And so on.

Originally, the two players had to get at least four question and answer pairs to line up. but requirement was lowered to two pairs a couple of episodes later. Four pairs seemed awfully unlikely, and I was kind of curious just what kind of chances the players had of surviving.

Thankfully, Mind Link conforms to a classic probability problem known as de Montmort’s Problem or the matching problem. The narrative I found the most was a bunch of mathematicians each go to a party with one hat. They all get drunk and leave, grabbing a random hat on their way out. What is the probability they end up with the hat they brought?

It’s like how it never occurs to the dining philosophers that they should just ask for more chopsticks.

But no one asks the important questions, like what arrangements did they make to get home that night? Surely none of them are driving if they’re too drunk to recognize their own hat! And why does no math lesson seem to present the scenario that I’m sure occurs to every student, where 𝑛 swinging couples attend a party and—

Anyway, I’m not going to reproduce the math here. There isn’t really anything I can add to the resources already available. Kyle Siergrist’s Random is succinct and thorough, but TC Jiang’s post made it all click for me.

The probability density function for the matching problem is:

\huge
\begin{align*}
  P(N_{n} = k) &= \displaystyle\frac{1}{k!} \sum_{j=0}^{n-k} \frac{(-1)^j}{j!}\\
  \text{where}&~n~\text{is the number of pairs}\\
  &~k~\text{is the number of matches}
\end{align*}

The only thing I can add here is a handy dandy interactive JavaScript calculator for the matching problem! Move the slider to change 𝑛.

The kind of neat thing is the numbers start to converge awfully quickly. The probability of 𝑘 = 1 is not very different if 𝑛 is 9 or 15 (and it converges to 1/𝑒).

So if our two players are given 10 questions, the chances they would get four question and answer pairs correct is just 0.37%! Even after the difficulty was reduced to two correct pairs, there’s still only a 8.03% chance they’ll succeed with their mind link. I think I’ll take my chances with that dead black mouth in the face of the great green devil instead.