title

The impossible chessboard puzzle

description

An information puzzle with an interesting twist
Solution on Stand-up Maths: https://youtu.be/as7Gkm7Y7h4
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: https://3b1b.co/chess-thanks
Home page: https://www.3blue1brown.com
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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
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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: https://github.com/3b1b/manim
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:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
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: http://3b1b.co/subscribe
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detail

{'title': 'The impossible chessboard puzzle', 'heatmap': [{'end': 68.056, 'start': 32.865, 'weight': 0.763}, {'end': 281.249, 'start': 257.153, 'weight': 1}, {'end': 821.344, 'start': 807.613, 'weight': 0.752}], 'summary': "Delves into the chessboard coin puzzle, exploring a prisoner scenario involving flipping coins to communicate a key's location, and discusses error correcting codes, highlighting hamming codes and strategies for the puzzle, considering 64^2^64 total possible strategies. it also discusses coloring cube strategies and the limitations for chessboard puzzles.", 'chapters': [{'end': 131.956, 'segs': [{'end': 32.265, 'src': 'embed', 'start': 4.139, 'weight': 0, 'content': [{'end': 6.58, 'text': 'walk alone into a room and you find a chessboard.', 'start': 4.139, 'duration': 2.441}, {'end': 10.062, 'text': 'Each of the 64 squares has a coin sitting on top of it.', 'start': 7.12, 'duration': 2.942}, {'end': 11.903, 'text': 'And taking a step back.', 'start': 10.822, 'duration': 1.081}, {'end': 19.686, 'text': 'this is going to be one of those classic prisoner puzzles where a strangely math-obsessed warden offers you and a fellow inmate a chance for freedom,', 'start': 11.903, 'duration': 7.783}, {'end': 23.128, 'text': "but only if the two of you solve some elaborate scheme that they've laid out.", 'start': 19.686, 'duration': 3.442}, {'end': 30.491, 'text': "In this case, what they've done is carefully turned over each of the coins to be heads or tails, according to whatever pattern they want it to be,", 'start': 23.928, 'duration': 6.563}, {'end': 32.265, 'text': 'And then they show you a key.', 'start': 31.164, 'duration': 1.101}], 'summary': 'Chessboard with 64 coins, prisoner puzzle with elaborate scheme and chance for freedom.', 'duration': 28.126, 'max_score': 4.139, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE4139.jpg'}, {'end': 68.056, 'src': 'heatmap', 'start': 32.865, 'weight': 0.763, 'content': [{'end': 35.687, 'text': 'They put that key inside one of the chessboard squares.', 'start': 32.865, 'duration': 2.822}, {'end': 38.488, 'text': 'Each square is a secret compartment or something like that.', 'start': 36.147, 'duration': 2.341}, {'end': 40.029, 'text': 'So you know where the key is.', 'start': 38.968, 'duration': 1.061}, {'end': 44.912, 'text': 'And the goal is going to be to get prisoner number two to also know where the key is.', 'start': 40.83, 'duration': 4.082}, {'end': 51.956, 'text': 'But the only thing that the warden allows you to do before you leave the room is to turn over one and only one of these coins.', 'start': 45.752, 'duration': 6.204}, {'end': 55.541, 'text': 'At that point you walk out.', 'start': 53.959, 'duration': 1.582}, {'end': 63.35, 'text': "your fellow prisoner walks in and with no information other than the set of heads and tails that they're looking at, which you've only barely tweaked,", 'start': 55.541, 'duration': 7.809}, {'end': 68.056, 'text': "they're supposed to deduce where the key is hidden, potentially winning freedom for the both of you.", 'start': 63.35, 'duration': 4.706}], 'summary': 'Prisoner puzzle: deduce key location with one coin flip.', 'duration': 35.191, 'max_score': 32.865, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE32865.jpg'}, {'end': 131.956, 'src': 'embed', 'start': 92.596, 'weight': 1, 'content': [{'end': 100.341, 'text': 'I remember the drive home was maybe three hours, and I think my mind was glued to the topic of flipping coins and encoding state that whole time.', 'start': 92.596, 'duration': 7.745}, {'end': 102.871, 'text': 'But the puzzle sticks with you even after that.', 'start': 101.089, 'duration': 1.782}, {'end': 106.754, 'text': 'After I solved it, I fell into these two surprisingly interesting rabbit holes.', 'start': 103.311, 'duration': 3.443}, {'end': 110.697, 'text': 'One was to prove that the challenge is actually impossible.', 'start': 107.434, 'duration': 3.263}, {'end': 116.442, 'text': 'if you vary the setup a little bit, maybe making it a 6x6 chessboard or maybe removing one of the squares.', 'start': 110.697, 'duration': 5.745}, {'end': 119.945, 'text': 'And to give you a little sense for where that rabbit hole leads.', 'start': 117.182, 'duration': 2.763}, {'end': 124.769, 'text': 'this video is going to end with an especially pleasing way to paint the corners of a four-dimensional cube.', 'start': 119.945, 'duration': 4.824}, {'end': 131.956, 'text': 'The other rabbit hole was to work out how closely you can connect the solution of this puzzle with error correction,', 'start': 125.814, 'duration': 6.142}], 'summary': 'Solving the puzzle led to two rabbit holes: proving impossibility and error correction.', 'duration': 39.36, 'max_score': 92.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE92596.jpg'}], 'start': 4.139, 'title': 'Chessboard coin puzzle', 'summary': 'Explores a prisoner puzzle scenario involving flipping coins on a chessboard to communicate the location of a key to a fellow inmate using limited information and strategizing, also delving into its impossibility in certain setups and its connection to error correction.', 'chapters': [{'end': 131.956, 'start': 4.139, 'title': 'Chessboard coin puzzle', 'summary': 'Explores a prisoner puzzle scenario involving flipping coins on a chessboard, with the goal of communicating the location of a key to a fellow inmate, using limited information and strategizing, while also delving into its impossibility in certain setups and its connection to error correction.', 'duration': 127.817, 'highlights': ['The puzzle involves flipping coins on a chessboard to communicate the location of a key to a fellow inmate, using limited information and strategizing.', "The exploration of the puzzle's impossibility in certain setups and its connection to error correction.", 'The puzzle sparked deep interest, leading to further exploration in proving its impossibility and its connection to error correction.']}], 'duration': 127.817, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE4139.jpg', 'highlights': ['The puzzle involves flipping coins on a chessboard to communicate the location of a key to a fellow inmate, using limited information and strategizing.', "The exploration of the puzzle's impossibility in certain setups and its connection to error correction.", 'The puzzle sparked deep interest, leading to further exploration in proving its impossibility and its connection to error correction.']}, {'end': 465.269, 'segs': [{'end': 169.668, 'src': 'embed', 'start': 146.5, 'weight': 0, 'content': [{'end': 156.323, 'text': 'So error correcting codes are a way to add a shockingly small amount of information to a message that makes it possible for the receiver to identify both when there is an error and,', 'start': 146.5, 'duration': 9.823}, {'end': 158.424, 'text': 'more impressively, precisely how to fix it.', 'start': 156.323, 'duration': 2.101}, {'end': 166.387, 'text': 'It turns out that the intuition for solving this puzzle is essentially the same as the intuition behind these things called Hamming Codes,', 'start': 159.204, 'duration': 7.183}, {'end': 169.668, 'text': 'which are one of the earliest examples of highly efficient error correction.', 'start': 166.387, 'duration': 3.281}], 'summary': 'Error correcting codes add minimal info for error identification and correction, with hamming codes being a prime example.', 'duration': 23.168, 'max_score': 146.5, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE146500.jpg'}, {'end': 250.755, 'src': 'embed', 'start': 224.214, 'weight': 2, 'content': [{'end': 231.458, 'text': 'Plus, it does more to help you appreciate the solution to the original puzzle when you can see how it is, in a sense, almost impossible.', 'start': 224.214, 'duration': 7.244}, {'end': 244.048, 'text': 'Where to start? What we want is some kind of visualization for what it even means to solve this puzzle.', 'start': 238.663, 'duration': 5.385}, {'end': 250.755, 'text': "And to build up to the general case, let's knock things down to the simplest case that we can that still has any kind of meaning to it.", 'start': 244.709, 'duration': 6.046}], 'summary': 'Visualize and simplify the puzzle-solving process to gain appreciation.', 'duration': 26.541, 'max_score': 224.214, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE224214.jpg'}, {'end': 283.231, 'src': 'heatmap', 'start': 257.153, 'weight': 1, 'content': [{'end': 261.178, 'text': 'One way that you could solve this is to simply let the second coin communicate where the key is.', 'start': 257.153, 'duration': 4.025}, {'end': 263.861, 'text': "If it's tails, it means the key is in the left square.", 'start': 261.598, 'duration': 2.263}, {'end': 266.484, 'text': "If it's heads, it means the key is in the right square.", 'start': 264.261, 'duration': 2.223}, {'end': 269.107, 'text': "Not a big deal, right? It's one bit of information.", 'start': 267.124, 'duration': 1.983}, {'end': 271.549, 'text': 'So when you need to change that coin, you can flip it.', 'start': 269.527, 'duration': 2.022}, {'end': 274.473, 'text': "But if you don't need to change it, you can just flip the other coin.", 'start': 271.85, 'duration': 2.623}, {'end': 281.249, 'text': "First things first, let's stop thinking about these as heads and tails and start thinking of them as ones and zeros.", 'start': 276.263, 'duration': 4.986}, {'end': 283.231, 'text': "That's much easier to do math with.", 'start': 281.589, 'duration': 1.642}], 'summary': 'Using a second coin to indicate key location, with 1 bit of information, easily changeable.', 'duration': 26.078, 'max_score': 257.153, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE257153.jpg'}, {'end': 478.925, 'src': 'embed', 'start': 446.379, 'weight': 4, 'content': [{'end': 452.702, 'text': 'With three choices for the color of each vertex and eight total vertices, we get 3 to the power 8.', 'start': 446.379, 'duration': 6.323}, {'end': 457.825, 'text': "Or, if you're comfortable letting your mind stray to the thought of painting a 64-dimensional cube.", 'start': 452.702, 'duration': 5.123}, {'end': 465.269, 'text': 'you can have fun thinking about the sense in which there are 64 to the 2 to the 64 total possible strategies for the original puzzle.', 'start': 457.825, 'duration': 7.444}, {'end': 469.131, 'text': "This is the size of the haystack when you're looking for the needle.", 'start': 465.909, 'duration': 3.222}, {'end': 478.925, 'text': 'Another attempt for the 3-square case might look like taking 0 times coin 0 plus 1 times coin 1 plus 2 times coin 2,', 'start': 470.501, 'duration': 8.424}], 'summary': 'Three choices for each of the eight vertices give 3^8 possibilities.', 'duration': 32.546, 'max_score': 446.379, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE446379.jpg'}], 'start': 131.956, 'title': 'Error correcting codes and puzzle strategies', 'summary': 'Discusses the importance of error correcting codes in information theory, highlighting the use of hamming codes. it also explores strategies for a puzzle, considering 3^8 or 64^2^64 total possible strategies for the original puzzle.', 'chapters': [{'end': 166.387, 'start': 131.956, 'title': 'Error correcting codes in information theory', 'summary': 'Discusses the importance of error correcting codes in computer science and information theory, emphasizing the ability to add a small amount of information to a message to detect and precisely fix errors, with hamming codes being a key example.', 'duration': 34.431, 'highlights': ['The intuition for error correcting codes is the same as that behind Hamming Codes, which are essential in computer science and information theory.', 'Error correcting codes enable the addition of a small amount of information to a message to identify and fix errors, contributing significantly to data accuracy and reliability.']}, {'end': 465.269, 'start': 166.387, 'title': 'Strategies for a puzzle', 'summary': 'Delves into the exploration of various strategies for a puzzle using visual and geometric representations, leading to the consideration of 3^8 or 64^2^64 total possible strategies for the original puzzle.', 'duration': 298.882, 'highlights': ['Visualizing the puzzle using geometric representations in higher dimensions', 'Total possible strategies calculation', 'Exploration of various strategies']}], 'duration': 333.313, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE131956.jpg', 'highlights': ['Error correcting codes enable adding info to fix errors, enhancing data accuracy.', 'The intuition for error correcting codes is the same as that behind Hamming Codes.', 'Exploration of various strategies for the puzzle using geometric representations.', 'Highlighting the use of Hamming Codes in computer science and information theory.', 'Total possible strategies calculation for the original puzzle is 3^8 or 64^2^64.']}, {'end': 1094.301, 'segs': [{'end': 571.05, 'src': 'embed', 'start': 503.74, 'weight': 0, 'content': [{'end': 512.337, 'text': "So it's like you're starting at the corner If you were to flip coin 0, that doesn't change the sum, so it takes you to another red corner.", 'start': 503.74, 'duration': 8.597}, {'end': 518.361, 'text': 'If you flipped coin 1, it increases the sum by 1, so it takes you to a green corner.', 'start': 513.119, 'duration': 5.242}, {'end': 523.946, 'text': 'And flipping coin 2 takes you up to 2, which looks like a blue corner.', 'start': 520.003, 'duration': 3.943}, {'end': 534.013, 'text': "The fact that you always have access to whichever color you want is a reflection of the fact that this strategy will always win if this is the corner that you're starting on.", 'start': 524.847, 'duration': 9.166}, {'end': 537.443, 'text': "On the other hand, let's say that you started at 0, 1, 0.", 'start': 535.023, 'duration': 2.42}, {'end': 543.892, 'text': "Well, in that case, flipping coin 0 takes you to another green corner, since it doesn't change the sum.", 'start': 537.445, 'duration': 6.447}, {'end': 549.737, 'text': 'But flipping either coin 1 or coin 2 happen to take you to a red corner.', 'start': 544.873, 'duration': 4.864}, {'end': 552.3, 'text': "There's simply no way to get to a blue corner.", 'start': 550.278, 'duration': 2.022}, {'end': 563.463, 'text': "Basically, what's happening here is that you have the options to subtract 1 by turning off coin 1 or to add 2 by turning on coin 2.", 'start': 554.414, 'duration': 9.049}, {'end': 566.686, 'text': "And if you're working mod 3, those are both actually the same operation.", 'start': 563.463, 'duration': 3.223}, {'end': 571.05, 'text': "But that means that there's no way to change the sum to be 2.", 'start': 567.707, 'duration': 3.343}], 'summary': 'Flipping coins to reach different colored corners based on sum and starting position.', 'duration': 67.31, 'max_score': 503.74, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE503740.jpg'}, {'end': 696.678, 'src': 'embed', 'start': 668.501, 'weight': 3, 'content': [{'end': 674.043, 'text': 'the problem where that came up was also phrased in terms of assigning colors to a high-dimensional cube.', 'start': 668.501, 'duration': 5.542}, {'end': 678.725, 'text': 'Though in that case colors were given to pairs of vertices instead of individual ones.', 'start': 674.744, 'duration': 3.981}, {'end': 684.848, 'text': 'The point is, analyzing how to color a high-dimensional cube is more of a transferable skill than you might expect.', 'start': 679.225, 'duration': 5.623}, {'end': 691.213, 'text': 'So to our question can you make it so that every vertex has a red, a green and a blue neighbor?', 'start': 685.987, 'duration': 5.226}, {'end': 696.678, 'text': 'Remember, this is the same thing as having an encoding for key locations,', 'start': 691.993, 'duration': 4.685}], 'summary': 'Analyzing how to color a high-dimensional cube is a transferable skill; can every vertex have a red, green, and blue neighbor?', 'duration': 28.177, 'max_score': 668.501, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE668501.jpg'}, {'end': 832.748, 'src': 'heatmap', 'start': 807.613, 'weight': 0.752, 'content': [{'end': 814.955, 'text': "On the other hand, every red corner is counted exactly three times, once for each instance where it's somebody's neighbor.", 'start': 807.613, 'duration': 7.342}, {'end': 819.343, 'text': 'So that final tally has to be three times the total number of red corners.', 'start': 815.782, 'duration': 3.561}, {'end': 821.344, 'text': "So, you know, it's simple.", 'start': 820.344, 'duration': 1}, {'end': 823.685, 'text': 'Find a coloring where eight-thirds of the corners are red.', 'start': 821.424, 'duration': 2.261}, {'end': 825.526, 'text': "Isn't that nice?", 'start': 824.925, 'duration': 0.601}, {'end': 832.748, 'text': 'Counting how many times some corner has a red neighbor is the same as counting how many times a red corner has some neighbor,', 'start': 825.946, 'duration': 6.802}], 'summary': 'Each red corner is counted 3 times, totaling 8/3 of all corners.', 'duration': 25.135, 'max_score': 807.613, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE807613.jpg'}, {'end': 974.974, 'src': 'embed', 'start': 948.122, 'weight': 4, 'content': [{'end': 955.144, 'text': 'Just by imagining, coloring the corners of a cube and then counting how many there are, you can conclude that no possible strategy,', 'start': 948.122, 'duration': 7.022}, {'end': 961.685, 'text': "no matter how clever you are, can work in all of the cases for this chessboard puzzle if the number of squares isn't a power of two.", 'start': 955.144, 'duration': 6.541}, {'end': 969.567, 'text': 'So even though it might seem to make it easier if you knock off a couple squares or reduce the size of the board, it actually makes the task hopeless.', 'start': 962.625, 'duration': 6.942}, {'end': 974.974, 'text': "It also means that the solution to this puzzle, which I'll point you to in a moment,", 'start': 970.512, 'duration': 4.462}], 'summary': "No strategy works if squares aren't power of two.", 'duration': 26.852, 'max_score': 948.122, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE948122.jpg'}, {'end': 1094.301, 'src': 'embed', 'start': 1075.396, 'weight': 5, 'content': [{'end': 1080.799, 'text': "And if you're curious about the connection with Hamming codes and error correction, I'm definitely game to make a video on that.", 'start': 1075.396, 'duration': 5.403}, {'end': 1082.019, 'text': 'just let me know in the comments.', 'start': 1080.799, 'duration': 1.22}, {'end': 1089.323, 'text': "I've been told that as far as motivating puzzles go, not everyone is as interested in symmetrical ways to paint a 64-dimensional cube as I am.", 'start': 1082.419, 'duration': 6.904}, {'end': 1094.301, 'text': "But reliable data transmission? Come on, I think we can all agree that that's universally sexy.", 'start': 1090.003, 'duration': 4.298}], 'summary': 'Hamming codes provide error correction for reliable data transmission, universally appealing.', 'duration': 18.905, 'max_score': 1075.396, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE1075396.jpg'}], 'start': 465.909, 'title': 'Coloring cube strategies', 'summary': 'Discusses a strategy for winning a game with 3 coins, analyzing the impact of initial coin settings on accessing different corners, and explores coloring high-dimensional cubes to represent key locations, demonstrating limitations for chessboard puzzles.', 'chapters': [{'end': 571.05, 'start': 465.909, 'title': 'Coloring the cube strategy', 'summary': 'Discusses a strategy for winning a game involving 3 coins and their sums, where a specific corner is always accessible, and the impact of initial coin settings on accessing different corners is analyzed.', 'duration': 105.141, 'highlights': ["The fact that you always have access to whichever color you want is a reflection of the fact that this strategy will always win if this is the corner that you're starting on. - This strategy guarantees a win if starting from a specific corner.", "But flipping either coin 1 or coin 2 happen to take you to a red corner. There's simply no way to get to a blue corner. - It is impossible to reach a specific corner from certain initial coin settings.", "And if you're working mod 3, those are both actually the same operation. But that means that there's no way to change the sum to be 2. - Certain operations in mod 3 lead to the inability to change the sum to 2."]}, {'end': 1094.301, 'start': 571.05, 'title': 'Coloring high-dimensional cubes', 'summary': 'Explores the strategy of coloring the corners of a high-dimensional cube to represent key locations, demonstrating that no strategy can work for chessboard puzzles with a number of squares that is not a power of two.', 'duration': 523.251, 'highlights': ['The chapter explores the strategy of coloring the corners of a high-dimensional cube to represent key locations.', 'Demonstrates that no strategy can work for chessboard puzzles with a number of squares that is not a power of two.', 'Discusses the connection of the puzzle to Hamming codes and error correction.']}], 'duration': 628.392, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/wTJI_WuZSwE/pics/wTJI_WuZSwE465909.jpg', 'highlights': ['This strategy guarantees a win if starting from a specific corner.', 'It is impossible to reach a specific corner from certain initial coin settings.', 'Certain operations in mod 3 lead to the inability to change the sum to 2.', 'The chapter explores the strategy of coloring the corners of a high-dimensional cube to represent key locations.', 'Demonstrates that no strategy can work for chessboard puzzles with a number of squares that is not a power of two.', 'Discusses the connection of the puzzle to Hamming codes and error correction.']}], 'highlights': ['Total possible strategies calculation for the original puzzle is 3^8 or 64^2^64.', 'The puzzle involves flipping coins on a chessboard to communicate the location of a key to a fellow inmate, using limited information and strategizing.', 'Error correcting codes enable adding info to fix errors, enhancing data accuracy.', "The exploration of the puzzle's impossibility in certain setups and its connection to error correction.", 'Highlighting the use of Hamming Codes in computer science and information theory.', 'The intuition for error correcting codes is the same as that behind Hamming Codes.', 'The chapter explores the strategy of coloring the corners of a high-dimensional cube to represent key locations.', 'Demonstrates that no strategy can work for chessboard puzzles with a number of squares that is not a power of two.', 'Certain operations in mod 3 lead to the inability to change the sum to 2.', 'This strategy guarantees a win if starting from a specific corner.']}