The Riddle That Seems Impossible Even If You Know The Answer

The 100 Prisoners Riddle feels completely impossible even once you know the answer. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription.

Special thanks to Destin of Smarter Every Day (https://ve42.co/SED), Toby of Tibees (https://ve42.co/Tibees), and Jabril of Jabrils (https://ve42.co/Jabrils) for taking the time to think about this mind bending riddle.

Huge thanks to Luke West for building plots and for his help with the math.
Huge thanks to Dr. Eugene Curtin and Dr. Max Warshauer for their great article on the problem and taking the time to help us understand it: https://ve42.co/CurtinWarshauer
Thanks to Dr. John Baez for his help with finding alternate ways to do the calculations.
Thanks to Simon Pampena for his input and analysis.

Other 100 Prisoners Riddle videos:
minutephysics:    • Solution to The Impossible Bet | The ...  
Vsauce2:    • The 100 Prisoners Puzzle  
Stand-up Maths:    • The unbelievable solution to the 100 ...  
TED-Ed:    • Can you solve the prisoner boxes ridd...  

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References:
Original paper: Gál, A., & Miltersen, P.B. (2003). The Cell Probe Complexity of Succinct Data Structures. BRICS, Department of Computer Science, University of Aarhus. All rights reserved. – https://ve42.co/GalMiltersen
Winkler, P. (2006). Seven Puzzles You Think You Must Not Have Heard Correctly. – https://ve42.co/Winkler2006
The 100 Prisoners Problem – https://ve42.co/100PWiki
Golomb, S. & Gaal, P. (1998). On the Number of Permutations on n Objects with Greatest Cycle Length k. Advances in Applied Mathematics, 20(1), 98-107. – https://ve42.co/Golomb1998
Lamb, E. (2012). Puzzling Prisoners Presented to Promote North America's Only Museum of Math. Observations, Scientific American. – https://ve42.co/Lamb2012
Permutations – https://ve42.co/PermutationsWiki
Probability that a random permutation of n elements has a cycle of length k greater than n/2, Math SE. – https://ve42.co/BaezProbSE
Counting Cycle Structures in Sn, Math SE. – https://ve42.co/CountCyclesSE
What is the distribution of cycle lengths in derangements? In particular, expected longest cycle, Math SE. – https://ve42.co/JorikiSE
The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1). – https://www.manim.community/

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Written by Derek Muller and Emily Zhang
Filmed by Derek Muller and Petr Lebedev
Animation by Ivy Tello and Jesús Rascón
Edited by Trenton Oliver
Additional video/photos supplied by Getty Images
Music from Epidemic Sound and Jonny Hyman
Thumbnail by Ignat Berbeci
Produced by Derek Muller, Petr Lebedev, and Emily Zhang