The impossible chessboard puzzle

An information puzzle with an interesting twist
Solution on Stand-up Maths:
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0:00 Introduction
3:58 Visualizing the two-square case
5:46 Visualizing the three-square case
12:19 Proof that it’s impossible
16:22 Explicit painting of the hypercube

Thanks to everyone who endured me probing them with this puzzle and provided helpful discussion, especially Cam Christensen, Matt Parker, and Mike Sklar. Mike, by the way, deserves credit for coming up with the particularly clean way to see that it’s impossible when n is not a power of 2.

These animations are largely made using manim, a scrappy open-source python library:

If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then “add subtitles/cc”. I really appreciate those who do this, as it helps make the lessons accessible to more people.


3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe:

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  1. Mmmm a way to do it is draw an arrow pointing the key. This doesn't work with small boards but works with every big enough board😂

  2. define turning. they all look look horizontally oriented to me – turn one 45deg in the board plane

  3. the warden sh*ts his pants while watching me balance the coin on its edge

  4. If me and the inmate were smart enough to understand this, we would't have ended up locked up in the first place.

  5. so was given a link to this for the solution off a clip and it flat says you wont' go over the solution. fastest way to get me to ignore a channel, pulling my leg.

  6. The Warden Only said "To flip a coin" not that you had to "put it back Where it was". As such, you could just Flip the coin and put it on the square where the key is.

  7. It's a coin, a 3d object. Why can't I simply balance the coin on its edge up against the wall, then let the warden recoil in horror at my brilliant use of physics while he sets us free?

    Don't give me the "that's not the right answer" nonsense because the right answer is the one that results in my freedom, not the one that is mathematically elegant.

  8. Although it is amazing how this makes sense, it makes you wonder.. who first thought of this ????

  9. "Solving" the puzzle before finishing the video to see if I'm even on the same page:

    I would tell my partner I'm going to flip every coin to heads. After, I'll flip the spot with the key to tails. GGEZ

  10. Could I move all of the pennys to the bottom right corner of their square aside from the key penny which would go top left?

  11. haha, i turn the t upside down

  12. You can solve any size board by refusing to flip a coin (a 3×1 board has 24 states where in only 1 state you need to refuse to flip one while a 2×2 board has 64 states where you need to refuse to flip in 8 states) the ability to refuse to flip gives 1 extra dimention of movement making it super easy to solve.

  13. chat gpt
    Interesting puzzle! Let's see if we can solve it together. To summarize the situation:

    There is a chessboard with 64 squares, each containing a coin. The coins are turned either heads or tails in a specific pattern determined by the Warden.
    The Warden hides a key inside one of the chessboard squares.
    You are allowed to flip over one and only one of the coins before leaving the room.
    Your fellow prisoner enters the room after you and must deduce the location of the key based on the pattern of heads and tails they see on the chessboard.

    To solve this puzzle, we need to find a strategy for flipping one coin that will allow the second prisoner to deduce the location of the key. Here's a strategy that works:

    Number the squares of the chessboard from 1 to 64, starting from the top-left corner and moving row by row.

    Count the number of coins showing heads. If it's an even number, do nothing (don't flip any coin). If it's an odd number, flip the coin on square 1 (top-left corner).


    When you enter the room, you have no information about the location of the key. Flipping the coin on square 1 will change the number of heads from even to odd or from odd to even.
    When your fellow prisoner enters the room, they will see the pattern of heads and tails. If the original pattern had an even number of heads, and you did not flip the coin, they will see an even number of heads and can deduce that the key is in the square that you flipped (square 1). If the original pattern had an odd number of heads, and you flipped the coin, they will see an odd number of heads and can deduce that the key is in the square that you didn't flip (not square 1).

    This strategy ensures that the second prisoner will have enough information to deduce the location of the key and potentially win freedom for both of you.

  14. Easier solution, I cough and sneeze all over the hand I use to flip the coin so when the other guy walks in he just has to look for the gross coin.

  15. If we can talk before it, I'll just say top left.

  16. Easy: set the flipped coin in the corner of the of the square it was set in instead of the center.

  17. Easy: set the flipped coin in the corner of the of the square it was set in instead of the center.

  18. Nope. I have a PhD and my mind is fried. Nope, this makes no sense to my mind

  19. As prisoner 1 take the key and walk out. Comrades are overrated.
    If you like your comrade: After walking away, file a case because of discrimination for the warden as he treated your comrade unequal.

  20. Me at 8:15: WTF there’s no freaking way.

    I’ll update later if anything changes

  21. Wait for the warden's reaction when I slowly steal all of the coins, declare myself as a pirate, and begin to march my pirate's song. After that, my fellow pirates will come and continue the marching, as the warden begin to scream in pure terror after hearing our ugly voice, then fainted after his ears bled. Even if that doesn't work, some of the prisoners and other wardens would notice our ugly voice. And when they came to our room, wanting to absolutely beat the sh*t out of the person who sang, I can just blame it to other pirates and the warden. They will do the rest, as I steal the key and escape 🙂

  22. I think I should know more math to understand this puzzle

  23. why cant you flip all coins to tails then the key space to heads indicating the key is there

  24. Hate the short you made for this but I must know now

  25. hi, here a solution i found (i hope) with a different approach

    1 = head, 0 = tails , example:

    1 0 1 0 1 1 0 1

    1 1 1 1 0 1 0 1

    1 0 1 1 0 1 1 1

    0 1 1 0 1 1 0 1

    0 1 1 1 0 1 0 1

    1 0 1 1 1 1 0 0

    1 0 0 1 0 1 1 1

    1 0 1 1 0 1 0 1

    this second matrix is just the chessboard with the corrisponding cell's number

    1 2 3 4 5 6 7 8

    9 10 11 12 13 14 15 16

    17 18 19 20 21 22 23 24

    25 26 27 28 29 30 31 32

    33 34 35 36 37 38 39 40

    41 42 43 44 45 46 47 48

    49 50 51 52 53 54 55 56

    57 58 59 60 61 62 63 64

    The third matrix is the first e the second multiplied element by element

    1 0 3 0 5 6 0 8 –> 23 sum of the row

    9 10 11 12 0 14 0 16 –> 72

    17 0 19 20 0 22 23 24 –>125

    0 26 27 0 29 30 0 32 –>144

    0 34 35 36 0 38 0 40 –>183

    41 0 43 44 45 46 0 0 –>219

    49 0 0 52 0 54 55 56 –>266

    57 0 59 60 0 62 0 64 –>302

    1334 TOTAL SUM

    ——– > 54 MOD,64 (mod of the total sum)

    The idea is to flip the right coin to change the rsult of the MOD in the number where the key is.

    for example, in this case you can flip the coin in position 21 or 43 (it depends if 0 or 1). after the fllp the result will be11. It is the cell number where the key is.

    so the prisoner 2 have to sum all the number in the second matrix, then he makes the mod 64 and he gets the position of the key.

    I tried some cases, and it looks like it is working, but probably I am missing something

  26. Plot twist: „Do meth“ is the message to one of the viewers though… 😅

  27. Bro, my friend is so dumb, that even if I told him where the key is, he will still fail.

  28. prisoner 2: guesses a 1/64 chance and gets it to solve the problem

  29. From the prisoner puzzle, the rule says you can flip one coin but doesn’t say that you have to flip a coin. So I am wondering, if “not flipping coin” is an option, I feel the 3d model will work since you can always stick too “key = the less number of head/tail” for example: 001,110 both tell the key is in the right-most slot. Of course. This is based on “if not flipping a coin is a possible option”

  30. Am I only one who is feeling urge to dislike and report this video

  31. Yes please make a video on relation to error correction codes

  32. I don't know anything about math but I can tell you that I can solve it by recording the coins around the key and tell him outside that the key lies in a place where it is vertically (0,1,1) and the horizontally (0,1,0) it is very rare that there woukd be common points but but if there are i would change them with my one move. And if there are multiple i would take the diagonals too.

  33. What else would people be discussing about in Grant's wedding?

  34. If the two “prisoners” so called can talk to each other’s beforehand they can come up with a pre-set cipher based on the chess board and they can actually solve it. However, if they cannot your proof applies. Thanks for the puzzle but no thanks for the solution. Imma go flip a coin.

  35. Me as prisonner 1
    Put the key under a square
    wait 1 month to get dust on every square
    flip the coin on the key's square
    Easy solution lol

  36. After finishing the video….. Couldn't thumbs down fast enough

  37. little does the warden know i’m a chess master and inliterally can just memorize the whole board

  38. all that for a "sorry, but your solution is in another castle" c'mon!!

  39. Why don’t you just flip over the coin with the key?

  40. 6:00 Easy at least to explain it when it comes to 3D world positions. Cube? Let's see! I just paused right now so let's see!

  41. It's not impossible, if you include decimals between positions in certain extent. x)

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