Why the 14-15 puzzle is impossible, and how to solve it anyway

[@dboyzero]

Does anyone sell an impossible version of the puzzle with the 15 and 14 already switched? Feels like it would be a fun prank to mix that up and have a friend give it a shot.

[@not-on-pizza]

On some 15-puzzles, you can carefully remove the tiles.

[@SimonClarkstone]

I've seen one with letters, where two of the letters were "E", and you needed to swap the "E"s over in order to get from the given stating postion to the right parity to solve the rest of the puzzle.

[@Nalkahn]

I have one... bought in the 80's ... The goal is to make a magic square with the numbers.

[@night.2257]

Wow there satan

[@mikmop]

If you turn the puzzle 90 degrees, you can then do a sideways solution to the puzzle, if you don't mind the numbers being flipped on their side.

[@Mr-dm5we]

I remember I had this toy and you could split the pieces from the base so sometimes I got a situation where 15 and 14 were swapped🧐. I thought I was a noob and didn't know it was literally imposible until now

[@leonardosanchezaguirre2459]

What do you mean with "This"

[@nekomimicatears]

@@leonardosanchezaguirre2459?

[@peppermintgal4302]

@@leonardosanchezaguirre2459 I think they mean that they had the toy in this video.

[@lqu]

oblivious genius

[@leonardosanchezaguirre2459]

@@peppermintgal4302 ok. didn't know that

[@neopalm2050]

The 15+4+4 puzzle is interesting in that we preserve even-ness again. It also only allows two orientations for each time.

[@梁偉康-d9k]

4gig

[@waldolemmer]

> evenness Aka parity

[@AstonCulf]

Rubik's cube has a few of these parity problems, with corner orientation, edge orientation, and piece permutation. The interesting thing about permutation parity is that you can have 2 corners swapped as long as there are also 2 edges swapped. As well as corner orientation which has 3 possible values which must equal a multiple of 3.

[@epsilonsolves1457]

I was thinking about this too. Wonder if there could be a “parity alg” to swap the pieces

[@tetrawaffle337]

There are parity algs on 4x4 and any larger even number cube, because edge orientation and permutation parity can actually happen in solves

[@AstonCulf]

@@tetrawaffle337 is this where I tell you that PLL parity isn't really parity because it's a 2e2e case

[@_..---]

Cool video man! I didn't even think this puzzle had any variants.

[@diribigal]

Henry was almost certainly the first person to design physical versions of the variants, and maybe the first person to conceive of these particular ones.

[@Minty_Meeo]

The 15+4 puzzle is a neat representation of a 2D hyperbolic space.

[@emy5845]

I would have put 13 14 15 16 together on the same panel, and 17 18 19 [empty] on the other. I wonder if you can switch from your setup to this one.

[@ferociousfeind8538]

This is a hyperbolic puzzle, where in some case, parallel lines do not exist. If we pretend we do, and count where to place tiles by following one line down the left side, and branching perpendicular lines off of the line to the right, at the point where five corners meet, the lines that were locally parallel are forced to diverge, creating a space. Here, we fill out the puzzle as normal, placing the empty tile next to 15, and then, as if in a panic, fit four extra tiles in the extra space that the negative(?) curvature grants us. It would, however, be neat if we got the 16 tile next to the 15 tile. And, given that the five corners meeting breaks the parity of the board, thats definitely a solution you could reach.

[@elderfrost9892]

the 15+4 seems at first glance and a bit of calculation that it would have parity issues and you couldn't get all arrangements. however, if you ignore the orientation of the board, and rotate every single piece around the center, so that each piece is rotated by 90 degrees, then solve with the peices in that orientation, would that possibly have different arrangements? not sure how to test it without the physical puzzle or a sinmulation, but it may be possible, as there is a odd number of pieces that take a odd number of moves around the center, odd x odd = odd. worth testing.

[@Vykori]

I mean the title says "Why the 14-15 puzzle is impossible" so I'm a little disappointed that explaining the concept of parity was supposedly beyond the scope of this video

[@henryseg]

Go for it!: en.wikipedia.org/wiki/Parity_of_a_permutation#Equivalence_of_the_two_definitions

[@Vykori]

@@henryseg thanks!

[@IsaacMyers1]

What about the 15-4 puzzle? It would be a cube corner you can flatten by breaking any edge.

[@henryseg]

I’ve thought a bit about what such a mechanism would be like - I think that a frame that can break apart like that (but not entirely fall apart) could be possible, although I don’t know how you’d keep the tiles from coming off the broken edge.

[@theapexsurvivor9538]

@@henryseg you could make it with 6 diagonal half squares that can fold behind it or open out to full squares that are solid to act as walls couldn't you? It'd still have a risk of falling out during the switch from one being open to another being open, or when all 3 were folded, but it should be fine when one is open.

[@henryseg]

@@theapexsurvivor9538 I’m not sure I’m following. Each full square could have a tile on it - presumably it would be a problem to fold one of them into a half square if there was a tile on it?

[@theapexsurvivor9538]

@@henryseg basically, there should be a total of 6 squares, 3 normal ones that can hold tiles, 3 split along with diagonal to allow folding that are solid and thus can't hold tiles. By unfolding one of the split squares you have 4 with only 3 usable, the fourth existing to act as a barrier to prevent the tiles falling off while you're moving them around. You can switch which one is open in order to rotate one of the tile squares 90° so that the two edges that bordered the previous split square are now in contact and tiles can be exchanged across them. Hopefully that makes a bit more sense.

[@henryseg]

​@@theapexsurvivor9538 Ah, interesting. I think I see what you're suggesting now. One difficulty might be that when you start folding a split square up in between two normal squares, it has to immediately start blocking movement of tiles across it. Also, weird things will likely happen when bending hinges if a tile is halfway between squares. There are some of these problems in the 15+4 puzzle as well.

[@Illogical.]

The same general concept applies to rubik's cubes and their variants.

[@not-on-pizza]

It feels like the 15+4 puzzle should have parity in its rotations, but actually, I'm pretty sure it doesn't. Consider the arrangement that you show in the video to be the canonical solution. If you ensure that the 1 tile doesn't ever move from its position (this is important, and I'll explain why in a moment), then by moving the other 18 tiles around (including around the central vertex), I believe that any arrangement that has the other 18 tiles in their correct locations must have an even number of rotations in all of the tiles. However, the 1 tile is not fixed in its position, and the frame's orientation is not marked in any way, so it's possible for you to move tiles partially around the central vertex in the same direction to reach a point that looks like it has an odd number of tile rotations have happened, even though all of the tiles are now on a different frame piece from when they started. But even if you mandate that each tile must be on the frame piece in the canonical solution, you can just move each tile completely around the central vertex (for the sake of argument, we assume that there is no limit to your patience), so that each tile is back in its canonical location, having rotated 5 times (and looking like just once). So... unless I've missed something, the 15+4 puzzle does not have parity.

[@neopalm2050]

Why must it be the case that any arrangement that has the other 18 tiles in their correct locations must have an even number of rotations in all of the tiles? Pick a special hinge. I believe it's possible to take the gap through the hinge clockwise once, then clockwise again but taking a different tile over the hinge, then never going through the hinge again and solving it back to the solved position as though it were a normal 2D 15-puzzle variant. This should take two different tiles to a 90 degree offset from their original orientation. I don't know for a fact if this is true for a 15+4-but-without-using-some-hinge but I believe any position with matching gap position parity and permutation parity is solvable.

[@ravi12346]

@ShrapnelCity I agree with your first paragraph, but I don't think rotating the frame makes any new positions possible. I think a general rule for whether a position can be solved is as follows. Call the edge between the 16/18 and 11/12 tiles "bad", and color the 20 positions with 0's and 1's in a checkerboard pattern, with the correct hole location on a 0, and with the pattern breaking (i.e. 0 touching 0, 1 touching 1) along the bad edge. Then any given configuration can be solved if and only if (checkerboard color of the hole) + (permutation parity including the hole) + (net number of 90-degree rotations) is even. Reason: any move that doesn't cross the bad edge flips the first and second parities, and any move that does cross the bad edge flips the second and third parities. So the sum of the three parities is conserved. Conversely, given a position which satisfies my criterion, you can fix orientations by carefully sending tiles around the center, and then solve it like a regular 15 puzzle. So exactly half of the 20! * 19^4 positions can be solved. The position where the entire puzzle is solved but rotated 1/5 of the way around relative to the frame *is* possible to solve: depending on whether you rotate clockwise or counterclockwise, this is either (checkerboard color 1, even permutation, 15 rotations) or (checkerboard color 0, even permutation, 16 rotations). (This confused me at first--it's not 19 rotations because the 3 or 4 tiles that move across the bad edge don't appear to have rotated.)

[@ravi12346]

@@neopalm2050 I read "even number of rotations in all of the tiles" as "even number of *total* rotations in all of the tiles", which is consistent with your example.

[@neopalm2050]

I've been trying to send a new reply saying I'd done it but youtube seems to be automatically deleting it or something.

[@neopalm2050]

There seems to be something wrong with the ending of the imgur link? I'll send it in small pieces or something? KZbin is being a real pain. htt hey youtube ps://img please stop ur.co deleting my comment, m/a/ag because there's absolutely pMS nothing wrong with it. NJ

[@bryanbischof4351]

Nice video. Im excited for who’s going to write up 15+4 first.

[@lucarioinfamous]

I remember trying for so long to solve the puzzle in descending order. It was one of the only ways I could never get it. *edit, just found out that it’s actually impossible. I guess that explains why I could never get it

[@emsytraut]

Same!

[@SlowerIsFaster139]

I could play with the 15 + 4 for hours. That was a really neat twist on the puzzle.

[@orisphera]

I know another version of the puzzle where any permutation is solvable. In that version, the tiles are at the vertices of an icosahedron. I have two ways of solving it. One way is as follows: until it's solved, pick the tile that should be where the empty space is or, if it's where it should be, any tile that is not in its place, and swap it with the current empty tile with 1 of 3 combinations depending on the distance between them. The other one is more like the way I solve the original puzzle. To make sure we're on the same page, here is the latter: For tiles 1-3, I just move them in a way that the empty tile, while going around the tile I'm moving, doesn't go over already placed tiles. That's impossible for tile 4. For the latter, I've seen people in ads moving the entire row, but I do this as follows: I move 3 into the place of 4 after making sure 4 doesn't take its place, then move 4 next to it, and then move 3 and 4 in their places. This detail, however, doesn't matter for making a way to solve the puzzle below. Then, I do the same for the second row. Then, I solve the third and fourth rows by columns. For the version I've described above, when I decide to use the second way, I first put the tile that should be the farthest from the empty tile to its place, then the tiles that should be adjacent to it, and then the rest. I'll write more details later

[@BradenBest]

I'd say it's still impossible because the underlying set theory still applies. It's just that now each object has two values attached (number and orientation). At the end of the day it's still a set, and just as it's impossible to turn (1, 2) into (2, 1) in an even number of swaps, it would still be impossible to reach half of the possible permutations in the modified puzzle. As for how to "solve anyway", I came up with a much simpler way: just target a known impossible permutation. If 0 represents the gap, then the states 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 and 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0 are respectively of odd and even parity. So if you have an impossible puzzle, just target the one with the gap on the top left and call that a solve. If you get to change the rules by adding a wormhole to the bottom right then I get to change the rules, too.

[@dodiswatchbobobo]

Every dang licensed video game when I was a kid… every single one had at least one of these things based around a picture. Usually it was a game mechanic.

[@drawacrab]

Ooo I love all these different variations! I’m reminded of what you can do with crocheting hyperbolic shapes

[@samuelwaller4924]

I made a simulator for this online and it somehow keeps being impossible, even though im scrambling with moves instead of just randomizing the position..

[@MichaelDarrow-tr1mn]

maybe you're interpreting the empty space as 0 and putting it in the top left

[@Amonfobious]

reminds me of how not all rubiks cube permutatiomsnare possible. if u take a solved cube and swap two pieces and mash it up you wont be able to solve it again, something will be out of place 99% of the time

[@MichaelDarrow-tr1mn]

actually it's 100% not 99%

[@brianthom5699]

Absolute magic. Thank you for this most interesting and informative presentation.

[@gryzor9967]

This is a perfect example of "what will happen when you're tired to find tutorial video after dislike counter got removed"

[@Iris-jw3ci]

I programmed a version of this game and didn't even know this was a possibility so a lot of the boards it generates are unsolvable

[@shermanfox7338]

So a old friend who moved away unexpectedly gave me one of those number things, still have it. Didn’t expect to find what it was from

[@hauntedsunsets]

it's really cool how the concept of a parity is already familiar to my thanks to rubik's cubes

[@nestor3540]

I just got that position and Im just glad it's Impossible, I was genuinely tweaking.

[@gerardodelacruz1430]

But how and why you get the 14-15 state from the initial position?

[@michaelkibble740]

Do you if older number puzzles is worth anything ?

[@Biltemaskorv]

Where can you buy just this one ?

[@CookedMeat]

All the smart guys are here for admiring the solution to the puzzles, meanwhile me is just here for the coo looking toys.

[@aukeholic1]

Can the 15+4 puzzle be purchased? It's really lovely

[@henryseg]

See the description of kzbin.info/www/bejne/fpSWqpmrjc6Km5I

[@keithortega9374]

When starting on lower left It is very easy coz I was used to it but But when Starting on the original format It always end in 15 14 number

[@summerblue6576]

0:36 I can only get mine to this position. I'm trying to find a video to get it 1 through 15😭

[@Nalkahn]

If you have the 14 and 15 switched, you are supposed to make a magic square with it, not put numbers in order ;)

[@MichaelDarrow-tr1mn]

@@Nalkahn or the empty goes on the top row

[@orisphera]

I have a proof of the parity principle that does not require one to flatten the permutation (i.e., turn it into a permutation of a finite number of numbers). For this proof, I use my own definition of permutation parity: the parity of a permutation is the parity of the number of loops in it if its length is even and the opposite parity otherwise. (Equivalently, the parity of a permutation is the parity of the number of loops of even length.) A swap either merges two loops or splits a loop; in both cases, the parity of the number of loops and therefore one of the permutation changes. Using this and the fact that every permutation can be made by swapping elements, I can prove that the parity of the composition of any number of permutations is the XOR of the parities of the original permutations and the parity of a permutation of a finite number of numbers is the same with both definitions

[@nightthemadqueen5499]

This is weird, because I can SWEAR me trying to solve this puzzle ALWAYS ends on 14-15 thing Better check that up

[@MichaelDarrow-tr1mn]

you might have one where the gap is supposed to be in the top left

[@michaeljayEFT]

I bought this same game, manufacturer that is. It's terrible where the pieces keep sticking. How can I fix this?

[@vj7248]

i'd go buy the qiyi one, since that's sliding blocks with magnets

[@lastplusfirst]

I somehow used to solve this regularly when I was a child but no longer can figure it out as an adult.

[@Dargonhuman]

I love these types of puzzles, and the easy way to solve them is to break them down into sections - there are several different methods but my preferred method is the home row method. How it works is you start with the top row and only focus on moving those tiles into position and ignore the positions of the rest of the tiles; they only exist to move the top row of tiles. After that, you move the rest of the left column into position without moving the top row and ignoring the positions of the rest of the tiles again. Then you solve the second row, then the second column and so on. What happens with this method is, with each row/column pair you solve, you're reducing the puzzle to smaller and easier boards until you get down to a simple 2x2. To use the 15 tile puzzle as an example: you start off with a 4x4 board, but after solving the first row/column pair, you're left with a 3x3 board, then when you solve the second pair, you're left with a 2x2 board. This method scales up to any size board that's playable with the only variation being how many reductive pairs you need to solve to get to the 2x2.

[@aashsyed1277]

Find a right triangle with all sides equal to a integer and its height is also a integer

[@hollyhart1715]

What’s the actual name of this ??

[@Van-Leo]

I love that KZbin is just like “here’s something you never knew about and it’s impossible, absolutely never, unobtainable!” “….Unless?” “Ok that was a lie, here’s the deal-“

[@sigmundfraud7068]

Holy shit I remember this puzzle from when I was like 10, I spent like a week straight working on it and I finally got it and it felt amazing this is so nostalgic

[@yesimew]

so is my puzzle wrong??? I solved everything except the last row, the end is 13 - 15 - 14 - blank sapce and i cant solve it! I dont remember if it was wrong when i bought it ;-; what a damn shame, ive spent so much time wondering why I couldn't do it???

[@MichaelDarrow-tr1mn]

i think the empty goes on the top row

[@j.vonhogen9650]

Great video! Thanks a lot!

[@77DC77]

_"The dots help you with the orientation of the numbers"_ Why is there a dot for 8 though?

[@henryseg]

You can turn 8 upside down and it looks like 8 again.

[@fabriziorasore5353]

It's there so that you can't confuse 8 with himself or infinity duh

[@chasemarangu]

An ending - good food for thought'

[@sarangmt]

I had this puzzle and was solving for last 14-15 switched... But some how I solve to get it back in line but here if I’m listening correctly he says it was impossible.. Now I doubt reality.

[@yuts568]

My dad and I have been stuck on this 14 to 15 numbers swopped around for two days🤣

[@VaradMahashabde]

Should've realized that the answer will be topological

[@khamphacongnghe247]

Does anyone played this game Slide Puzzle Camera?

[@baco5963]

I solved this puzzle a lot of times when i was like 6 yo, now i dont even know how a child solved this

[@chimetimepaprika]

Jajajajajaja when you pilled out the burrito puzzle it was so unexpected amd hilarious. Jajaja

[@danielbrown7064]

Do you have the 3d files available for printing myself? thnaks!

[@ashton2283]

Title: impossible. Also title: ez

[@JoBot__]

So you can solve the puzzle by changing it to be a 2D spherical world or a 2D hyperbolic world. Cool. ^-^

[@sachinkumar-jc3ub]

really a good video

[@Mcnoklacyphie]

Be patient, is a lot of sandwiches, and 1/2 of life (90 years a life) to solve…. Ahhhhh I’m so pissed

[@MRCAB]

Very cool.

[@xyznihall]

great video

[@Spiffy.Griffy]

cubers that know 4x4 will find this funny: just do OLL parody

[@lorenzodepaoli3642]

lol a friend of mine just offered me the puzzle and I solved it in less than 10 minutes withot having ever even tried it

[@orisphera]

Why 14-15 is impossible: 14

[@Dreams_Of_Lavender]

15+4 puzzle = hyperbolic 15 puzzle

[@IxodesPersulcatus]

Certainly an odd way of doing that.

[@electricmaster23]

Sam didn't actually invent this. A cursory reading of the Wiki page will show why.

[@NeuroPulse]

Very cool!

[@PixelatedPixelz]

I have this exact one! I love doing it in my spare time and it is really fun. It can be challenging sometimes but I end up completing it in 5 or less minutes USUALLY. I do it 1-15 but never tried 14-15. I should try it though! Also, I find it interesting that it is impossible to do 15-1 aka backwards. I try but never succeed, does anyone know if it is in fact possible?

[@MichaelDarrow-tr1mn]

if two squares are swapped from how you want it, it's not possible

[@yuklungleung620]

just group theory

[@So-Be-It_890]

1879

[@meowmeowmeowmeowmeowmeowmeowml]

Here before this blows up

[@Cxntrxl]

so the solution is just a fuckin wormhole xP

[@Dziaji]

HOLY CRAP! That first second of the video is jarring!

[@Stillminator]

I do not learn anything, i just practice and some under than mins but 8 puzzles, my highest is 6 secs

[@davegreen9155]

I love the MATH you show....and how you ACCEPT that such things can "CHANGE",.... But I wonder....is "MATH" the lesson. Or is SACRIFICE? 'SCIENCE" gas many of tge SAME "RULES" as almost anything else..... (Yes....it IS a "PUZZLE"😎) But "WE" have the very BAD habit of "NEEDING" to be "SIGNIFICANT" which is WHY we literally CREATED everything we had to overcome😁 _yeah I have to face this too so....GLAD I am not alone

[@emanuel3596]

Me readign the title: what

[@emanuel3596]

Me rading my comment: what

[@emanuel3596]

Me reading my response: what

[@cybersilver5816]

@@emanuel3596 this was a wild ride, thank you.

[@emanuel3596]

@@cybersilver5816 np?

[@ScienceDrummer]

I think Patrick could solve it.

[@백과사전-p5l]

좌빨우파봤는데 이게 나오네? 알고리즘 무엇 ㄷㄷㄷㄷ

[@ebrucan7161]

i solved it in 6 minutes... its very easy

[@SOJITZL]

Wait... ???????¿¿¿¿

[@realase387]

care bout ur ugliness first, then the maths i cant imagine i live in the society where its ok to look like this

[@WillyTheMuffin]

I see you felt threathened by his intelligence and had to pick from the lowest hanging fruit to make yourself feel better

[@nicreven]

that is so unnecessarily mean :(