I thought it would be best to always take from the sides and place in the middle. Then I remembered that the game has winners and losers and that the objective isn't just to stack the tower as tall as possible without it falling over.

Imagine your opponent takes out a block, and then you premove checkmate

And then there's Jenga with a chess clock: each player has one minute in total. Strategically no difference, but toppling is way more likely and it gets more fun as you have to hurry all the time.

Analyzing jenga like nim is like solving every problem with a hammer

Really nice video! A lot of people commented on problems regarding real world Jenga. But in my opinion they missed the point, i.e. the beauty of mathematical abstraction. You did a great job of breaking the game down to its essentials. The messy real life problems should not be of our concern in such an abstract analysis.

Nice video, now I want to see Jenga analysed from an engineering perspective.

Jenga blocks are not identical. They're casually "identical" from a glance. Every Jenga set I have bought there are two thicknesses there are hard to distinguish from each other. The thinner blocks result in the very easy blocks that you can remove. A good setup randomly distributes the thicknesses.

I have an extremely hard time thinking of jenga as math and not physics.

to answer one of the end questions, an intuitive answer for why the table repeats every 3 levels would be a modulo 3 argument, that is that 3 layers provides the perfect amount to "throwaway" moves to return the game to one of the initial states in starter 9x9 grid. That is any move outside of the final 3 full layers transposes into one optimally

Great video, well done on the animation quality

For the 4 block jenga : the strategy of the second player is to take the block on the same level as the first player so that it’s impossible to remove any more blocks on this level. The second player will always have a legal response move to play, and it will eventually win.

I feel like this needs to outright define a tower "falling without collapsing" as "still falling". Because I've seen videos where a player could and did remove the last block in a level (usually by hitting it from the side really quickly), causing the tower to technically fall but in such a stable way that it stays intact (essentially, yanking a table cloth from underneath a flower/candle display and leaving it standing). And if people in reality can accomplish this, players with idealized abilities certainly can, which would make almost any game infinitely long.

Real (easy to imagine) applications of advanced concepts are great to watch and learn, here is your 9th subscriber.

you have great things ahead of you with how nice this video is

In the first example, if you are accounting for the stability of the tower, if you take one of the side blocks and put it in the position that is over the other side block, then the tower will “fall” while your opponent is taking the only valid legal move. For this reason, there are only 4 moves that lose the simple example, not 6 losing moves.

I guess no moves like “pinch the outer two blocks together then remove one” I would rather call P-positions “Z-positions”, Z stands for Zugzwang

This video is awesome! Keep it up!! Now I know how to memorize and play an optimal game of combinatorial-ideal Jenga!

The reason why the second player always wins in 4-block-Jenga: There's always the possibility to remove 2 blocks from each level. While 3-block-Jenga has variations of either one or two blocks being removed per level.

It’s really strange that in analyzing a game about falling there is no use of physics… in the two block version if you took a block from the side and placed it on the opposite side from where you removed it then your opponent taking either the side or middle block would almost certainly cause it to topple.

You forgot one last assumption that you made. All the blocks are the same size... perfectly. In real Jenga not all the blocks are the same size but they're close. This forces the middle one to be structurally essential sometimes as it's taller than the other two and the other two literally aren't doing anything. This is why some people can blow on a tower and remove a Jenga piece.

By the way, actual jenga blocks have slight variations in thickness so that some become easier to remove in certain configurations.

Put it in my watch later because the topic is so incredibly interesting!!

Every time I play Jenga, I am constantly distracted by thoughts of "There should be a mathematical solution/strategy which tells you what moves to make to guarantee victory, and I really need to sit down one day and actually spend time thinking about it and solving it." Thank you for figuring out the logic so I don't have to. I will probably have a hard time memorizing the solution grid like you mentioned, but at least I won't be distracted with thoughts regretting not working it out as now I don't need to and can instead refer to this video.

Thank you! Very interesting!

But for the simple example shown at 4:06 the top left and bottom right options also guarantee an immediate win, since the opponent is only able to take the opposite side block, at which point the centre of mass is no longer above the remaining base block.

Love the video and ideas

36k views for your first video seems pretty nuts!! Keep up the good work!

As an aside, while it wouldn't have been good to cram into this video, it's interesting to consider the game of combinatorial Jenga with multiple towers (so you lose when you're forced to topple one of them because you have no good move to make). That's something that's almost as easy as the original problem if you have tools from Combinatorial Game Theory, and probably seemingly hopeless if you don't.

This is incredible. Nice job! I'm totally subbed now :)

This was great!

Great video! Don't mean to add too much pressure, but I'm looking forward to seeing more of your videos in the future! You've got amazing editing skills. Consider making some minidocs about stuff you're interested in ;)

Great video, finally I now know the rules of Jenga and I know how to win it from a combinatorial point of view. Thanks.

This channel about to go ham

In the description you say you remove gravity, which of course, drastically affects a real Jenga game :) but cool video!

Excellent animations and description. I was expecting to see way more subscribers.

Great vid! Hope you'll turn up the volume for your future vids though :)

Great content, thanks

Rainbow Tumbling Tower lid has been my weed tray for 15 years.

great video!

Very impressive video ❤

very cool

Great vid man, i thought u had at least a few thousand subscribers

Just when you thought it couldn't be better

The 2-level analysis is for Jenga newbs. Pros will shift a side block to the center to enable taking a block meaning whoever goes first loses every time.

Let's call this game "Jena", since it requires zero g to work! (and also no friction between pieces, but I can't make a pun with that)

Great video! Now what I'd be interested is a follow-up that tells you the best move based on the current situation. Optimally condensed in a shape where you don't have to carry around a lookup-table. So, if we're in an N-position: How can we classify the cases of moves to a p-position? If we're in a P-Position: how can we maximize the probablility that an opponent who doesn't know the strategies, leaves us with an N position in the next turn? And, are there easy mnemonics? So, if we start with the table at 11:20, and shrink it down to a 9x9 table ( or more likely a single column and an 8x9 table, because of the symmetries), instead of labelling the points, N and P, label the N-Positions with the corresponding move(s) to a P-Position, and the P-positions with the corresponding moves to "those N-Positions that have most N-positions to go to"? With any luck there will be choices, that might be made in a way that the table has a symmetry that will only require to remember, for example, 3 entires of the first column and a 4x3 matrix, then all you'd need is the column mod 4 and the row mod 3?

good video can’t believe there are so few subscribers, So now you have 8

Analysis for two players add the complexity of alternating turns and strategy. I'm also interested in max number of moves, rather than "which player will win".

Nice video! I think your table may have an off-by-one error. Note that the first column of the table in your video doesn't follow the nice pattern of the rest of the grid. I reproduce your table if the first column allows f = -1/3 (so the second column is the real f=0), etc. (As if you could, say, win a game by taking the other side piece of a half layer to complete the untouchable top layer (which isn't counted in f).)

Mathematician's Jenga strategy in shambles when they realize there's a legal technique to pull blocks from "impossible" layers. I can one-handedly stabilize the tower while flicking out a one-block row.

cool video!!!

Interestingly you can also stack 5 jenga blocks in a level if you rotate them 90 degrees. Would be interesting to know what would happen if you were allowed to choose the orientation when starting a new top level.

Can those tabletrick moves be included, where the last piece in a layer gets flicked out?

The volume on your videos is way to low. I had to almost double the volume compared to other videos or even the ads on your video.

Technically, you could launch rest of the tower up, and have all of the peices long ways up.

You would also have to assume you live in a world without physics. You can make an unbalanced jenga tower while following the jenga rules by taking pieces only off of one side. That way, you can end up taking a piece that will cause the tower to fall. If you can avoid taking that piece, you can force your opponent to take that piece. If you can do that then the strategy changes.

Long ago, the four nations lived together in harmony Then everything changed when the FRACTIONs attacked

You left us hanging! Does the standard 54 block game start in an n or a p position?! I was watching till the end to get this answer!

You forgot the method of quickly swiping a lone middle block from its level

What about redistributing the weight of the tower to one side by always adding to it?

nice

3:30 I don't understand? Is the top completed layer not legal to remove blocks from?

Very interesting. I wonder how this would go with Jenga Xtreme, where the pieces are slanted and the game can end prematurely.

Why does the table look like it repeats after 3 rows? What am I missing? Cause a 3x9 table sounds way more reasonable than 9x9.

Aren’t there ways to guarantee the structure collapsing other than just moving a middle + side block from one row, though? If a side block is removed from floor 1, and then the *opposite* side block is removed from every parallel floor above (I.e. all the odd numbered floors), doesn’t that move the moment of inertia (sorry if that’s the wrong term) of the structure?

put this in a physics simulator and take all the left side blocks, and always put them left-first in the topmost layer. see what happens, then tell me if the side matters or not.

What about stability? The tower will buckle faster, if there are a lot of middle stones. There is a formula F= E*I*pi^2/(ß*L²), which limits the height. Placing a stone on the top will also decrease the limit height, if the weight is not centric.

Seems like there is another unspoken assumption, which is that we have to ignore center of gravity considerations in extreme/contrived arrangements of the blocks, no? E.g. if you remove a base-level outer block, but all the blocks on the similarly-aligned layers above are removed on the opposite side (perhaps with intervening layers only having the center block, if necessary), the COG would tip the tower (even with perfect placement), yeah?

your jenga analysis only works well with a perfectly modeled jenga. a real life jenga has shape irregularities that block you from having the side block/middle block choice in some cases, unless you are skilled enough to remove such a blocked piece without endangering the structural integrity of the tower. but a really skilled player can sometimes also remove a sole middle piece, effectively removing a layer from the tower without the tower falling. that means in a perfect setup with perfectly skilled players, the only way to guaranty victory is by intentionally taking the right amount of middle pieces out of the tower to unsure your are the player with the last viable turn. well, you were only talking about the math aspect of jenga, so your model is still correct. and the reason why a four block jenga is always a win for the second player is easy, there are no single middle pieces and therefor each layer has always two pieces to remove with no possible choice to switch between P and N positions. this is true not just for four block jenga but for any even number sized jenga.

It's not that hard to remove a block from a double, it just requires a balanced base since the goal is to shuffle the tower above a single block. Makes it way harder to solve since if you play by the one touch rule you would be able to lock out sections based on stability. Pretty sure the pattern looping every 9 moves is due to that being the limit before a player is able to force it back into a winning starting state, but proving that is mainly busywork so it's just a guess.

wish the final perfect game was shown in 3d. if they tried to make the game last long as possible, would it be just all single block layers?

It's definitely possible to take out all three blocks in one of the sections, if you do it just right you can get it to fall down a block's height without collapsing. Is that a legal move in either case?

I have a major issue with this analysis. In your "perfect" game of jenga, the tower never falls over, and a player can only win by making it so the other olayer has no remaining moves. However, you failed to account for the physics of gravity and center of mass. For example, if both players take blocks from exclusively one side of the tower, the tower will begin leaning to the opposite side, even if all blocks are perfectly placed and perfectly the same size. This will cause differences in which blocks are bearing the load and change the game entirely, even in your "perfect" jenga game. I suggest running a physics simulation and apply your "perfect game" to see what really happens.

4:01 cant the opponent take one of the blocks below the newly placed one or am i missing something?

wouldnt -- -- O O O O O O -- fall on the slightest disturbance when picking the only option as the com would shift slightly towards the edge of the block placed previously? i know we have perfect steady ness but even random air currents would make that fall if the blocks are perfectly uniform.

I'm confused at 3:33. Why can your opponent not make any moves? "You can only pick from he topmost complete level", so you pick from the 2nd layer and move one to the top layer? But then again, I don't play Jenga so might not understand what the rules are...

11:29 Neat visual artifact. Probably an editing mistake? Other than that, a perfect video. A very good analytical look at Rainbow Jumbling Tower!

Yeah 4 Block per level Jenga would be pretty obvious how to win for the second player because every level would no longer have a middle block, and thus every level is always like the case where you have a pair of blocks. You either have the Middle two or Outer two blocks, and removing any more blocks would make the tower unstable. Thus if the first player removes a Middle block, you also remove a middle block to clear that row. And if they remove an Outer block you remove the other outer block to clear that row. Tho even then I don't think you need to follow that strategy to the letter to win. If Player 1 removes a block in one row and then both players decide to move onto another row instead, there will always be that row that you can go back to to set player 1 back on a losing strategy. And if player 1 decides to remove the other piece in that row, then you can remove the other piece in whatever row you just touched and we're back to square one. So in fact it's not just trivial for Player 2 to win in 4 block jenga, but I think it may actually be *impossible* for Player 1 to win under optimal circumstances. there will always be an even number of valid moves no matter what with no way to change the possible states of a given row. So since the last player to make a valid move will always win, then Player 2 will always be the last one to make a valid move.

What happens when each level has 4 blocks? It depends on your assumptions. I'm going to make the assumption that, given the blocks in a layer are named [1, 2, 3, 4] in that order from left to right, the layer will topple if blocks 1 and 2 are removed, or if blocks 3 and 4 are removed. In this case, no matter how you play, exactly two blocks can be removed from a layer before it becomes "locked" and never again has any moves. The game starts with every layer having two moves left on it. Player one picks any of them arbitrarily, and player two can simply deplete the other move from the same layer*. Eventually they will run out of layers after an even number of moves, thus being in a P-position with P1 to move. *Actually, it doesn't matter what order the moves are done in. The game will always, without fail, take an even number of moves. P2 literally cannot lose, unless they purposefully move a block that topples the tower. But what if we tweak the assumption? We're already considering some forms of perfection, why not extend that to manufacturing perfection? Lets say every block is *exactly* the same dimensions, down to infinite precision, and that the tower is constructed with every block perfectly flush with every adjacent block. In this case, removing blocks 1 and 2 from a layer still leave the exact midpoint of the layer above supported, and perhaps that's enough. This means we assume that a layer will topple if blocks 1 and 2 and 3 are removed, or if blocks 2 and 3 and 4 are removed. This is quite a different game, and I don't know immediately how the game state can evolve. Presumably we could do what you demonstrated around 9:16 ; simplifying the position to a count of layer types and filling in a grid. But this time, there are four layer types; filled layers, layers with one side removed, layers with one center removed, and layers with two blocks removed excluding the case where both centers are removed (plus the fifth case of a "dead" layer with no moves left). I am currently too lazy to proceed with this any farther; the rest I leave as an exercise to the reader.

Its cool that you described a version of Jenga with perfect players but why arent the players considering tower stability? Like imagine the scenario where the bottom layer is missing its leftmost block and all the upper layers are only missing their right most block then the tower would fall. Like you would also need to keep track of how a move would shift the center of gravity of all layers and if that shift would move the center of gravity into an empty space.

woah! what did you use to animate it?

This is an interesting explanation of the strategy of Jenga, however it does not inform a strategy for the real world game. That does not make it invalid but I want to point at something. The assumption that players play perfectly steady is not sufficient for the described turn of events, the assumption that the blocks are frictionless is a must. See, if the blocks have friction you can create an invalid move that is making the tower tumble due to the friction of the piece being pulled far enough from a incomplete level creating a moment. This does not apply for small towers, for an arbitrarily lower friction you can create an arbitrarily tall tower that will tumble. Not even to say the irregular sized pieces. For the creator of the video: Don't take the criticism badly, you made a wonderful video exploring an ingenious way of using mathematics to solve a problem and you did it thoroughly. However sometimes it's hard to relate with the thrill of solving such problems and people can be let down thinking of more down to earth problems. You done what you set up to do and that's awesome.

And then I play and the human factor comes into full force and I knock over the tower although it would have been a winning position for me. 😂

Why does the pattern repeat every three jenga levels? I think it is because every three levels guarantee the creation of an additional level during gameplay, and that somehow resets the table to the 2-,3-, and 4-level states.

I'm not sure why people are arguing over physics. This is clearly a game theory approach looking for mix strategy under a hypothetical setting… how jenga blocks are manufactured and how the tower develops are totally irrelevant… makes you wonder just how much education the average commenter has on YT. Sheesh

10:46 why (1 , 2/3) is not a N position, since you can move to ( 0,2) that is a P position

The definition of the ordered pair at 8:55 was really unclear and it made the rest of the video incomprehensible to me. What does (13 and 1/3, 3) mean?

What s the strange picture at 11:29 ?

quick maths

You also assume there is no air resistance.

Your subscriber count is kind of insanely low. I'd keep it up if I were you. Interesting ass video!

this is the type of guy that people don't want to play with :D

Really great start! I can tell you're taking this seriously, so I'll give you my thoughts on how to improve. Keep in mind I'm not a mathematician or a youtuber, just a random guy, so I might be off on some things. The first thing that stood out to me was the audio - it's not great. I'm listening on my headphones and I have everything turned up, and the dialogue was still a bit quiet, and the mixing felt like it was drowning the dialogue out with the music on top of that. Breaking up the video into different sections with different music tracks could help with the pacing, because only having a couple Mozart tracks playing in the background made everything feel like it all blended together. The Jenga concept is great for a math video, but it felt like you were ignoring the physics part of it. For example, if you start by removing the bottom left or bottom right block, and keep stacking more blocks on top, it would eventually topple, right? I think it would. It seemed like you were ignoring weight distribution except in the case where the middle block was removed, and that's not usually how Jenga games play out in real life. It would have been very interesting and added a whole new level of complexity if you factored this in. It would have been a fun and satisfying ending to the video to watch 2 bots play out a game (or many games) of Jenga following the ruleset you'd found was best, and I was sad that it didn't happen. Alternatively, you could have filmed yourself playing by the ruleset with a friend in real life to see how well it played out. That's not necessarily in the spirit of pure mathematics, but it would definitely be an entertaining ending. Also I have a nitpick. Rather, a pet peeve of mine. At 12:00 you talk about how this has all been proved by dry inductive arguments, but 'surely a more intuitive proof exists.' Why is that a sure thing? Math doesn't exist to conform to our ideas of simplicity or elegance, and it's entirely possible that there aren't any more intuitive proofs for Jenga. It's entirely possible that there are, but saying 'surely' rubs me the wrong way. Again, I'm not a mathematician, so maybe you have better insight here, but if you do, you should expand on it in the video rather than just assuming. I took a look at your profile and it looks like this is your first video, and it's very impressive! Keep it up!

My only problem is balance. At 6:44 you remove a side block and lay it straight ontop because your assumption says that nothing you do ontop matters, i say that assumption is false and ruins the integrity of the science. I say that very move you make is a unstable one that couldnt be done, you creating a gap and than lay your only weight straight ontop of that gap. The weight of that move is the same on both sides but the balance is unfavorable. You have a thicker side to a longer side. Like a L with a fat foot, that foot wants to go down. I like your video, i love using math to see whats possible in a game. But i really believe the assumption that you can lay your blocks however you want ontop ruined the integrity of your test. Because that simply isnt true, this move would make the top 2 levels come down and you would lose. Cool vid though.

But IRL the blocks r identical + get damaged over time losing pieces

You misspelled partisan

Wait… Taking blocks from the topmost complete layer is not allowed? Really? 😮 … Actually, come to think about it… Jenga has *rules*?

The audio is very low, I can't hear a thing you said, am I the only one ?

This was interesting, now do chess next 😂

So the first person always wins in jenga? So unfair