The Past Speaks

Kai P. Kaufmann is a PhD student here at the Management School of the University of Liverpool.

The puzzle he designed inverts certain assumptions of game theory by requiring cooperation to succeed and has been praised by several public intellectuals.

Here is the puzzle. 

N persons are kidnapped by aliens. If they solve this puzzle, they are made board members of the intergalactic council; otherwise our planet will be colonized: The panopticon is a ring-like building with N+1 empty, soundproof cells, each of which has two (shared) automatic doors that give access to the neighboring cells. Walls, doors, ground and ceiling are white and solid. On each door, there are two light bulbs (one on each side of the door). The N individuals are placed one by one inside the first N cells and instructed:

• (a) Knowing the number N, they have O(N³) minutes to walk through the N+1 cells and get back to their original cell, proceeding from 1 to 2, from 2 to 3, …, from N to N+1, from N+1 to 1.
• (b) Sitting in their cells, they receive a signal that indicates the beginning of the countdown and that consists of all door light bulbs being turned on nearly simultaneously.
• (c) Person 2 may not rise before person 1, person 3 may not rise before person 2, …, person N may not rise before person N-1.
• (d) Initially, all doors are closed. When someone presses the automatic door opener, his/her neighbor may not be inside that cell.
• (e) One minute before the end of the countdown, a light signal of the same kind as before will be given.
• (f) When the time is up, person 1 needs to sit in cell 1, person 2 in cell 2, …, person N in cell N.
• (g) Person 2 may not sit down before person 1, person 3 may not sit down before person 2, …, person N may not sit down before person N-1.

To prevent boredom, an infinitely stretched Nelly Furtado song is played. After the experiment, they get their previously collected time measuring/communicative devices back and are returned to Earth.

With the fate of mankind at stake, describe how the humans got out of the panopticon to win the second prize of £100, and explain the reasoning of the panopticon builders to win the first prize of £1000. (Side comment: Answers to both questions exist, are well-defined and unique.)

Send your answers to kaufmann@liverpool.ac.uk.

You can get more information about this contest here. 

Motivating statements

 “Many thanks, Kai, for sharing that intriguing puzzle.”
Steve Pinker (Harvard)

“thanks! i’m at a conference now … i’ll chat about this when i see
magdalena.”
Dave Chalmers (Australian National University, NYU)

“Thanks! I won’t even try – I’m not a natural born mathematician; lack
patience to become even a half-decent chess player, and didn’t even
get the second layer right on Rubik’s cube. Would Tower of Hanoi lead
in the right direction?”
Per Davidsson (Queensland University of Technology, Australia)

“Thanks Kai for sharing”
Michael I. Norton (Harvard Business School)