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 get 20% off a yearly subscription.
Special thanks to Destin of Smarter Every Day ( , Toby of Tibees ( , and Jabril of 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:
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:
Stand-up Maths:
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. -
Winkler, P. (2006). Seven Puzzles You Think You Must Not Have Heard Correctly. -
The 100 Prisoners Problem -
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. -
Lamb, E. (2012). Puzzling Prisoners Presented to Promote North America's Only Museum of Math. Observations, Scientific American. -
Permutations -
Probability that a random permutation of n elements has a cycle of length k greater than n/2, Math SE. -
Counting Cycle Structures in Sn, Math SE. -
What is the distribution of cycle lengths in derangements? In particular, expected longest cycle, Math SE. -
The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1). -
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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
Produced by Derek Muller, Petr Lebedev, and Emily Zhang

Пікірлер: 27 624

  • pyguy

    Something seems wrong at

  • WetBadger

    When you factor in the odds of one nerd convincing 99 other convicts to go with this strategy, your chances quickly fall back to zero.

  • DooDooBear
    DooDooBear21 күн бұрын

    It would be awesome to someone like Mr. Beast actually do the experiment with people to show it being done.

  • Michael Texeira
    Michael Texeira

    The minute you mentioned loops, it no longer seemed impossible and actually seemed retroactively obvious. Math is so damned cool.

  • Vimal Kumar
    Vimal Kumar

    the question at

  • herlocksholmes1888

    My Chemistry teacher was OBSESSED with this channel in his high school years (he's pretty young, yeah), and I just had to check it out for myself. I've always had a bad relationship with Math because most of my teachers were douchebags, but these videos are helping me see it through a new perspective. It's still frustrating and confusing at times, but it's oddly cool nowadays. It gets less tiresome to calculate stuff now. Thanks awfully, dude! 😄

  • TheJulez

    What people don't seem to get is, that you choose the box with your number, so you can only come back to it by finding your number on a letter. If you start the loop on your number, it can only end on your number.

  • Dr. DJ X
    Dr. DJ X

    As somebody that's tried tracking down a CD left in the wrong CD case, I can attest that the loop strategy does indeed work 31% of the time. (The other 69% of the time it turns up weeks later on the kitchen table.)

  • Matthew Haff
    Matthew Haff13 сағат бұрын

    I don’t consider myself very proficient with math or numbers really, but somehow this creator makes it fascinating. Well done, Sir! 😊

  • Tyler Peacock
    Tyler Peacock

    To me, the neatest thing about this is that by definition, getting on the loop with your own number will ensure that unless you are on a loop of 100 numbers, there are going to be boxes that you never have to open because they aren’t on your loop. It doesn’t tell you by how much your odds increase, but you necessarily make you own pool of options smaller unless all are on one loop.

  • Ben Wright
    Ben Wright

    As soon as I was presented with this puzzle, my first thought was to find a way to make the individual prisoners' chances of success correlate with each other (didn't actually find the trick by myself, though). Once you break independence between the guesses, you start making gains in overall success. It'd be interesting to see how the limit changes as you alter the proportion of the boxes each prisoner is allowed to open.

  • Thomas Rosenberger
    Thomas Rosenberger

    I actually think that this is quite intuitive once you hear it. I didn't know it before, but as soon as I heard the following the loop part of it everything made sense.

  • Aaron S
    Aaron S

    Interesting corollary: If prisoner #1 (or any other prisoner) finds that his own loop has a length of exactly 50, he immediately knows there's a 100% chance of success.

  • Karthik Menta
    Karthik Menta

    The most important part of coming up with a solution (if you didn't know the riddle solution already) was always how do you constrain the actions of your fellow prisoners so they open the boxes systematically. This is important - by opening up the boxes systematically you can attempt to remove randomness.

  • stetson nesbit
    stetson nesbit

    I feel like the question “how can you guarantee you’re on the right loop,” is people actually thinking you can guarantee it in the 50 moves. You’re absolutely guaranteed to find your number in 100 moves, and that’s what you’re saying here, you’re not guaranteeing that the prisoners will be freed. But no box or slip will send you to a box or slip you’ve already opened.

  • Gabzsy

    This channel is amazing. It reinvigorates my interest and curiosity about rational thinking and mathematics that my father instilled in me decades ago. I didn't really ended up practicing those muscles in my life but they are still there.

  • Mr. Mayor
    Mr. Mayor

    You need to do this with a group of 100 people to see if it works. I won’t believe it until I see it. Math is like magic.

  • Caleb Whitehead
    Caleb Whitehead13 сағат бұрын

    I actually had to pause the video and just sit back because my mind was so blown. I don't remember a video ever having this kind of effect on me. Very well explained. Thank you

  • Greg Squires
    Greg Squires

    I think the chance of convincing 99 other prisoners that this strategy is their best chance of survival is much lower than 31%.

  • Jordan Cooper
    Jordan Cooper

    Wow this is so clever, I wish I thought of this at the beginning! In hindsight it seems more obvious, knowing that the only relationship between all the prisoners is their number they are originally assigned. The answer which probably hits everyone is that the only option is to be random which will obviously not bode well, but it seems the only other possible option would be this loop method. Any algebraic manipulation of the loop method would produce either the same result or worse and I can't seem to think of any other logic method which (aside from the loop method and random) that would actually produce any meaningful search method i.e. reasonable win strategy. If anyone else does have another search strategy that is not the two above mentioned I would be very interested to hear!