Orbifolds and Other Games

A Labyrinth of Lies

Fri 03 July 2020 by Moshe Zadka

In the 1986 movie Labyrinth, a young girl (played by Jennifer Connelly) is faced with a dilemma. The adorable Jim Henson puppets explain to her that one guard always lies, and one guard always tells the truth. She needs to figure out which door leads to the castle at the center of the eponymous Labyrinth, and which one to certain death (dun-dun-dun!).

I decided that like any reasonable movie watcher, I need to implement this in Python.

First, I implemented two guards: one who always tells the truth, and one who always lies. The guards know who they are, and what the doors are, but can only answer True or False.

import dataclasses
from typing import List

guards = [None, None]
doors = ["certain death", "castle"]

@dataclasses.dataclass
class Guard:
    _truth_teller: bool
    _guards: List
    _doors: List[str]

    def ask(self, question):
        answer = bool(question(self, self._guards, self._doors))
        if not self._truth_teller:
            answer = not answer
        return answer

guards[0] = Guard(True, guards, doors)
guards[1] = Guard(False, guards, doors)

This being a children’s movie, the girl defeats all odds and figures out what to ask the guard: “would he (points to the other guard) tell me that this (points to the door on the left) door leads to the castle?”

def question(guard, guards, doors):
    other_guard, = (candidate for candidate in guards if candidate != guard)
    def other_question(ignored, guards, doors):
        return doors[0] == "castle"
    return other_guard.ask(other_question)

What would the truth-teller answer?

guards[0].ask(question)

True

And the liar?

guards[1].ask(question)

True

No matter who she asks, now she can count on a lie. After a short exposition, she confidently walks through the other door. It’s a piece of cake!

Thanks to Rina Arstain and Veronica Hanus for their feedback on an earlier draft. Thanks to Glyph Lefkowitz for the idea to use dataclasses. All mistakes and issues that remain are my responsibility.

Some related work in the field of formalizing logic puzzles: