Parity Puzzles
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- Опубликовано: 9 фев 2025
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“There have to be an even number of gears...”
The Möbius gear loop from Matt Parker's video: May I introduce another dimension
But then what if you have 3d gears instead of 2d gears
@@richardpike8748 4d loop. ezpz
@@cobaltbeau you mean bottle right?
@@anawesomepet you mean kleinbottle, right?
For the marble puzzle (right after the gears) another way to look at it is that the marbles are corners of a triangular piece of paper, and each time you pass a marble through, you're flipping the paper. To get the paper facing the same way, you need an even number of flips, so you can't with 15 because it's odd
i thought of it as moving the same marble back and forth and youd need an even to get it to its original position
He didn't say 15, he said 'after 15' and what comes after 15? 16, which is a even amount. So it is possible
@@ForgottenFafnir Shitty bait
A video where the thing on the thumbnail is instantly adressed and explained. Like I really am getting what I was looking for. *WOW!*
0:55
Actually, there is a way to do it: exploiting the third dimension.
If you turn this loop into a mobius strip it works perfectly fine.
pentagon [ˈpen(t)əˌɡän]
NOUN
A plane figure with five straight sides and five angles. •Since in ‘2D’ they don’t count 5 inside corners plus 5 outside corners =10 corners, we assume its 5 straight sides are already in a Möbius-arrangement, and when at every angle is a single gear, one of those 5 gears is considered inverted by 180° to rotate it in the opposite direction. It is now trivial to suppose that the existing rule still must apply, as the described quintet cannot turn.
@M Detlef wow
Whoa, this escalated quickly.
Watching avengers endgame
cough cough Matt Parker cough cough
Just rotate all the gears around the center. Problem solved.
Listen here you little shit
Genius. The "No" solution was trivial.
Just either add another gear or remove one.
_I'm gonna do what's called a pro-gamer move_
That was my thought straight away, for me it was not a case if it mechanically worked but the answer lay in the way the question was asked as you point out
“There would have to be an even number of gears”
Me: ZeRO gEaRs
Zero is even though
@@themobiusfunction I know
Great new video Zach! :)
Wow, no replies for 7 months. Love your vids, Flammy.
Wow, no replies for 6 months.
@@anawesomepet wow, no replies for 3 hours
@@greedo69 wow, no replies for 3 weeks
@@Osama-Bon-Jovi-01 hi sam
If it weren't for your work, i would have never expanded the horizons of my curiosity and would have never been aware about such possibilities of mathematics/science
Thank you so much🙂🙂🙂😊
Great work 👍👍👍
no u
Ijjjjjjjj
your gears in the animation dont mesh when rotating.
gg ez
Elsa’s hair didn’t mesh with her shoulder, either.
i hope people know that rooks can traverse more than 1 square at once
Welp, Zach Star definitely doesn't know this... :/ He kind of gets all those puzzles wrong.
Oh yeah i actually went to open an editing app to do this and i managed to solve it, maybe he should've used a different chess piece to show that puzzle
Had to replay this bit multiple times, but one rule is "you can't go over a already visited cell" which eliminates the possibilities we have
@@gregoirepelegrin1966 Starting the puzzle by skipping non-visited cells works fine.
In the chess puzzle, you can't put a rook there as it can move as much tiles as it wants
The rule said that you can't touch square that you have already traced over, meaning moving to another square removes every square in between
So basically, you can only move 1 square each
But if you can jump multiple tiles, there would be no parity and its possible
5:45-5:50 definitely thought he was doing the equivalent of solving a Rubik’s Cube by peeling off the stickers and putting them back on lol
An 8 Puzzle IS a twisty puzzle. Any cubers here?
@@wildrubikslegokids1242 I'm a cuber. Twisty puzzles twist. This does not twist.
Me, an intellectual: *One Gear.*
1:00 oh we’re assuming the gears can’t enter the 3rd dimension well then of course not
They can all rotate at the same time, but it's only for a very short time as they will almost immediately bind.
5:38 Ok, but your explanation made it crystal clear! And you always bring impressive content 🔥
When I clicked the video I thought you were gonna explain that it is impossible because there is always some clearance between the teeth of two neighbouring gears.
This channel deserves more subscriber. The content quality is so premium
This makes me think of a region in a game I've been playing lately called "HyperRogue", which deals with hyperbolic geometry. There's a region called the "Warped Coast" that uses a different tiling pattern than other regions and has an enemy type that only moves when I act, they'll wait if I wait, so parity becomes an issue if an incoming enemy is on the same tile shape as me. Luckily, there's trees dotting the land areas that I can chop down to "act" and thus change my parity!
The rook one is not worded very well. It's legal for a rook to jump around, further clarification is needed as to what "goes over" means in this scenario.
3:17 Rooks can actually move more than one square, so the colors don't necessarily switch
No, but every single move can be expressed as a sequence of one-square moves. That's what he's analyzing
That moving accross one square still technically included touching a square
@@somebodyiusedtoknow2012 In that case, you don't need 63 1-square moves so it is possible
Like all the squares that it skipped over it technically "touched"
Yeah, better if he used a king for the puzzle. Also, I didn't undestand if a starting square counts or not.
* sees title *
you fool, I have watched all matt parker's humble pi presentations
for the number puzzle I just used my rubik's cube intuition and said no, since you can't have two pieces swapped either
Zach, you might like puzzles involving Lagrange's Theorem. A really interesting fact is that a finite product of finite cyclic groups is cyclic (can be made by "rotation" of a single element) if and only if all the orders (size) of each group are relatively prime. This explains why you can't generate all the possible states of a typical combination lock with repeated applications of the same move. For example, rotating the first notch twice and all the others once. You'll always be missing something.
I don't understand what you're saying but sounds cool
That IS a really interesting fact.
Would love to see a video on that!
Rooks dont need to move to tha adjacent tile
These gears always remind of HR's basic graphic of team work where it shows 3 locked gears. Perfectly on point illustration.
It IS possible to link an odd number of gears and have them rotate but it's 3D. Tilt each gear a little more each time so that you get a sort of moebius ring of gears.
for the polygon example, I thought about the line that every time it intersects with a side it switches from exterior of the polygon, and since it starts and ends in the exterior it has to cross an even number of sides
3:38 the rook can move further in one move.... It isn't the king. Which can only move 1 in any direction. Rooks can move 1-8 any vertical or horizontal as long as its on the board and doesn't come across another piece
Correct me if I’m wrong- but isn’t the 8x8 rook puzzle possible because a rook doesn’t have to move to an adjacent tile? I think I’ve found a potential solution using that rule, but there isn’t one if the rook does have to move to an adjacent one.
Holup, rooks can move more than one square, meaning it can go from white to while in one turn. Unless you consider all the tiles it went over in one turn touched.
100% true
but that would still count as 1 turn, meaning you can move 2 tiles per turn and breaking the puzzle
Just use r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2 to solve the parity problem.
No Rw U2 x Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw'
A rook can move more than one square at a time.
That doesn't matter. Moving 2 squares, 3 squares, or even 7 squares "at once" is materially identical in this case to moving over the squares 1 at a time: there's no opponent to countermove, and moving "over" a square counts just the same as stopping on it.
Another cool parity problem is the problem of whether it is possible to play with a Rubik's cube until it is solved except for a single edge piece being flipped. Kinda takes a lot to introduce the concept of parity of a permutation though.
3:10 Technically a rook could do it. A king, not moving diagonally on the other hand, could not.
@Logan Post Yeah you're right. I didn't think it through when I wrote it.
@Logan Post It matters how you understand the problem. If you consider a multiple space move to touch all of the sq uares that the rook has passed, then it is indeed impossible. Otherwise there is an easy counterexample
@Logan Post Not if you lift the rook to move it. Or if you're online, you could always just disable animations if your chess server has that feature. :P
The first rook problem is solvable. The error in the logic was that every rook move goes from white to black squares.
If the first move, the rook castles (king side). The rook goes from a square to another square of the same colour. The rook does not touch the square in between. The other 62 moves are as normal. So you end on a white square and can end in the opposite colour.
There's many ways to do it. The flaw in this video is he states that "the rook will change from white to black each move" however a rook can move as many spaces vertically or horizontally in a single movement, therefore skipping the "on/off" problem posed in this video. In the end, allowing you to complete the task with relative ease.
Me, a cuber, reading the title: 4x4? 5x5?
Yeah!
Those are even parity but ok
Rooks can move from a white square to another white square (or black to black) on consecutive moves, they don't have to move one square at a time. To solve the puzzle, You can go up and down one square at a time From A8 until G3, then go to G1, then G2, then H2, H3, H4, H5, H6, H7, H8 and finish on H1.
The rook don't have to switch colors every move, but a knight must
In an actual chess game, no, it doesn't.
But, in the puzzle, it does.
Moving it 2 squares is the same as moving it one square in the same direction twice. Therefore, the difference is trivial.
The rules for this puzzle weren't explained properly.
@@Leyrann except its not, if the rock skils a square, it later can come back to that square.
@ゴゴ Joji Joestar ゴゴ but passing through ≠ landing there
You can think of the marbles as going forwards, and then the same moves in reverse, thus even.
If you have an odd parity, you can rotate the gears to form a Möbius strip it would work fine
I immediately got the polygon with a line crossing all sides. I feel so smart!
7:26 now there are 3 out of order pairs, the 2 and 0 are out of order in the top area
rooks are not limited to moving one square horizontally or vertically, therefore one can move from a black square to a black square
>First example is a gear problem
Matt Parker approves
Here’s a fun one: let’s say you are playing tetris, but every piece you get is one of those J shaped pieces. Prove that if you want to completely clear the board, you have to clear some number of lines which is a multiple of 4. Show that perfectly clearing after 10 rows for instance, is impossible.
The chess one works if you go all the way right, go down to the square right above “end”, go left 1 to the white square and take it up 5 to the black square . Then go all the way left to black square. Go down 1 to white then go right 5 to black square. Now take that down 4 to black square then left one to white, up 3 to black then over all the way to black. Go down 1 to white then right 3 to black. Down 2 to black left one to white, up2 to white, left 2 to white. Down one to black, right 1 to white, down 1 to black left 1 to white, down 1 to black then slide over to end
Show me you know Dynamic Programming without telling me you know Dynamic Programming : This Video
Rubik's cubes are also another example of this concept
Me: opens video
No
Me: Closes video
For the 1-8 puzzle, I kept finding things that were invariant when two numbers were swapped
Well, I mean, they won't all rotate individually if the teeth are all linked, but they could all rotate as a system
2:40 is possible, because the rook can skip over some of the squares and then go back to them later.
2:27 just do it on a mobius strip lmao
Actually in mobius loop you have to have odd number of gears for it to work (because of half twist rotation gets reversed as you go around)
tbh i was actually waiting for a video on this topic....and its unbelivable that here it is....thanks @Zach Star
you're welcome
If even then yes, if odd then no.
NO TWO ADJACENT GEARS MOVE IN SAME DIRECTION. ONE IS CLOCKWISE NEXT WILL ROTATE ANTICLOCKWISE.
Fun fact. There's a coin or something or other about unity...
It has 13 gears
There should be a joke video on this, called "Parody" Puzzles.
I'll show myself out now.
LMFAO
and dont come back
The problem is this: I don’t need a 12 minute video to answer a yes or no question...
It's the first time I skipped the video to see the sponsor part.
Let's assume the gears are made from stainless steel. Therefore sound (and thus deformation) travels at roughly 5800 m/s in them
What if we now put a row of gears around the equator, so that the first gear can rotate for 20,000km÷5.8km/s = ~57 min before the both different turning directions "collide" on the opposite side of the earth
Or does the rotation propagate faster than the speed of sound in the material?
hmm I am looking at the chess problem and unless I am going something wrong I have an example of the rook ending on the white square. It is the fact that rooks can move backwards. So from the start in the video. The solution is to make a snake shape on the two up rows exiting to the right third row in which you will now start on a black square so end on a white square now using linear moves only.
I thought the title said *Party Puzzle* and I'm like, "who'd listen to this at a party..?" 😂
"if your mom is larger than the universe, is she the universe?"
-some guy idk prolly my brain
No. It must be an odd number inbetween. An even number of gears.
Initial guess looking at the first few seconds only: If you move a cog clockwise, tracing it around an odd number of cogs will mean it should also move anti-clockwise, which is a contradiction.
...
Cool. That was easy! Now for the rest of the video.
Marble triangle::
Even= Clockwise == ABC (or BCA or CAB)
Odd = Anti = ACB (etc).
Any of the marble motions toggles the clockwise/anti-clockwise parity. So again; odd numbers of moves will always change the parity of a marvel arrangement, so 15 moves cannot get back to the original.
I’ll try the rook thing after my meeting!
64-1 = 63 is odd, so toggling from one colour to the adjacent one (not diagonal) will swap colours from white to black or black to white. So 63 toggles from white will leave you on a black spot , which is not the goal. So the task is impossible.
For the 5 by 5 version, the value 25-1 is even. So the parity proof does not show impossibility. In fact you can easily just start at the top left, go all the way down, one to the right, all the way up, etc. Snaking you’re way to the bottom right. So this version is solvable
I checked, and:
1) The rook on the 5x5 can complete the route from all white squares.
2) It is possible to perform the single line cutting with any even number of edges (although, of course, not with every shape)
Ziv Ronen so you’re going to have us all believe 1) you spoke with The rook, and 2) you also speak with cousin It.
I'm noticing how he pronounced polygon. First syllable sounds like "poll"/pole.
I learned to say it with first syllable rhyming with wall/Paul/doll/call/mall
i paused to try and figure out the rook puzzle only to find out the rook can only move one square at a time
U wrote parity in heading... that helped me to solve in seconds.
You are wrong about the rook one. Just go all the way to the edge in a straight line, then move to the side by one, then repeat until you get to the end.
I firgured the answer before even clicking on the video
edit: k now I found out the video is not only about that
Similar thing happens when you rotate 2 pieces of a rubics cube and you can still solve it
Does anyone know what kind of animation software he uses ? it looks pretty cool.
The ABC puzzle I didn't think about as clockwise/counterclockwise, but instead as flipping front to back along one edge. So after an odd number, you have the formation upside down.
Yes they can *rotates the surface the gears are on*
I count 11 gears I am still in the adds at the beginning, didn't count the teeth yet. But my gut feeling is it will not because it isn't an even number and it only has one rotation axis. If you add another axis you could get an odd number of ⚙️ to rotate I'm working on a 5 Axis motor at the moment just for fun and adding a 40° twist between axis. Stabilizing the magnetic field is a pain though, I'm not sure I've found true center for the arrangement yet. 😳 Also keeping perminant magnets from flipping and or shooting out of the center to one pole. I'm probably going to try a 5 phase 10pole next and just scrap this idea for lack of headache.
Rubik's cubers waiting for their moment
Wrong question in the a-b-c puzzle :). "Can you get the original configuration after 15 moves?" Definitely yes at move number 16, 18, 20, ... But can you get it in exactly 15 moves? No.
I enjoyed this a lot!
Having an odd number would be possible in the last puzzle if you use a piece of paper and make a tube out of it.
That would make it mobius strip-like. In the first puzzle - with the cogs - if you use a mobius configuration, you CAN actually get an odd number of cogs to work. There's a Numberphile video on it somewhere.
@@Leyrann it's a standupmaths video you're talking about
The chessboard one got me thinking of Newtonian Paths
The third one was nothing to me because of those Ted-Ed riddles
I got all of these right but half of them for the wrong reasons
Just started the book Mathematical Circles: A Russian experience an hour ago and the gear problem was the first problem there!
Yeah it's a great book, exposes you to a wide variety of contest problems and ideas. Quite rigourous as well. Btw are you in high school, doing olympiad math?
@@manswind3417 I was in my drop year at the time. Was solving it out of interest, but also to try to get into CMI. Didn't get in though haha
@@aarjavjogani Oh okay apparently I'm 6 months late in asking it lol. Never mind though, the book is quite useful and irrespective of where you are, you can certainly succeed. Atb to you!
@@manswind3417 haha yes, it was a lot of fun as well! I've completed around 80% of the book. Are you in college or something?
No, 11 gears can't rotate because 11 is an odd number.
The disappointment him not showing all gears moving..
11 gears, no
It had to be even
I thought you were going to show PARTY puzzles !! :)
Obviously not drinking games !
The answer to the chess puzzle is - yes.
A rook can move to any tile is a straight line, Not just to the adjacent ones.
You do realize that moving N squares in a straight line is the same as moving to an adjacent square N times, right? The same reasoning applies whether you consider full rook moves or 1-square rook moves.
@@zealiskander, no it is not. You can choose which tiles you touch. So if you moved from E1 to E8 for example that leaves E2 to E7 untouched. This way you can leave the desired tile for your last move.
@@Maks_Liadetskyi listen to the video. “You cannot go over tiles you already traced over”.
Going from e2 to e7 uses up e3, e4, e5, e6, and e7.
@@zealiskander, well that makes no sense.
The question is: "Can a rook move around such that it Touches each square only once and end up on the bottom rigth tile?". The answer is Yes, because you can jump from tile to tile leaving some in between untouched. Why couldn't you jump over tiles you already traced over without touching them?
If the guy in the video meant what you mean, the question schould have been: "Can a rook move around such that it touches each square only once and end up on the bottom rigth tile moving only from one tile to an adjacent one?" or use a king without diagonal moves.
The one in the thumbnail was so easy lmao
The answer to the question posed in the thumbnail is no because there is an odd number of rotating gears in the interconnected sequence, which each is being made to rotate in one of two directions of rotation opposite its neighbours, thus clashing when its odd neighbour rotates in the same direction as the one it's being made to reverse.
You asked.
It was explained in the first minute of the video. But I didn't come here to watch; I came here to answer the question in the thumbnail.
It was explained in the first minute of the video. But I didn't come here to watch; I came here to answer the question in the thumbnail.
YOU DIDN'T DRAW THE ROOK PROPERLY!!!!
U can Draw line around halfs of the gears... If they meet ... You can roste them ať the same time .. if not you cannot
8:00 so basically you are thinking about a 2D game as if it were a 1D game
I remember the first one from Mathematical Circles by Dmitri Fomin
Can't rooks move more than one tile at a time?
I remember Numberfile did a video several years ago
3:20 no it doesnt rook can move more than 1 square at a time