# Puzzle similar to 15-puzzle but more moveable pieces?

In the name of Science!™, I'm trying to look for (or, in the worst case, create) a puzzle/game similar to the n-puzzle whereby a player or player(s) have to put some pieces in a particular order (e.g. in natural order in the case where the pieces are numbers) in the fewest number of moves, and there are restrictions on the possible moves of the pieces which are dependent on the positions of the other pieces on the board (e.g. in the case of the traditional n-puzzle, a piece can be moved to an adjacent position not already occupied by another piece).

# Reducing likelihood of physical adjacency without increasing branching factor (much)

However, in the game I'm looking for, I need the movement capabilities of each piece to be less restricted than in the traditional n-puzzle: In the n-puzzle, there are at most four possible pieces which can move at any one time (because there is typically only one empty space and pieces can only move horizontally or vertically). Moreover, during much of the game, there are fewer than four moveable pieces, so the average branching factor of the game is actually quite low.

## Physical location of pieces vs. movement

Unfortunately, e.g. allowing diagonal moves means that the game is too easy, and the moveable pieces are still all in the same physical area on the board, which I'm trying to avoid, anyway: I need to find a way to keep the moveable pieces from being "near" (i.e. adjacent to) each other on the board, so you can't just think in terms of e.g. "move the piece

• to the left of
• to the right of
• above
• below

the free space" — namely, you have to consciously identify the pieces without depending on their locative relationship with other pieces (the pieces also lack any obvious unique identifiers such as by being numbered: they just have abstract shapes on them, which are not necessarily unique). At the same time, the player(s) must be able to move exactly one piece at a time (no groups of pieces in one turn).

Are there any "sorting" puzzles/games which can easily be played in this way (for minimizing the amount of moves) and which still have a less "predictable" set of moveable pieces? If not, how might I go about making one? Off the top of my head, I'm thinking it might be useful to allow pieces to "jump" over each other similar to in checkers but without capturing each other — would that be a useful way to increase the number?

• A Rubik's Cube comes to mind. – Angzuril Oct 27 '16 at 14:56
• Perhaps backspin fits the bill? – Will Oct 27 '16 at 15:35
• – Rand al'Thor Oct 28 '16 at 15:16

As answered by @BeastlyGerbil, you have the world of twisty puzzles. Here in the Twisty Puzzle Museum you can find over 5,000 of these kind of puzzles, and here is my personal collection of currently 279 puzzles (pictures are a bit outdated though, since I now have a few more shelves; list is up-to-date however).

That being said, you gave the following comment on Beastly's answer:

Yes, that is interesting, but the bad part about them is that you move multiple pieces at a time, and I need a game where you only move one at a time.

So perhaps you are looking for something like:

The MoZhi Rainbow 4x4x4 Cube, which is basically a 3D Fifteen Puzzle:

Of which there are also different sized versions, like the 2x2x2 (Inside Out 2x2x2 Start / Vadasz Kocka 2x2x2) or 3x3x3 (Peter's Black Hole Cube).

Something else could be the Wisdom Ball - BREAKTHROUGH, where you have one hole and fix rotating disks to move all the pieces one by one:

Bolaris - Vastaväri, where you again have one hole (at the bottom of this picture), and you can move all the tiles around one by one:

Or my own homemade Beheeyem Octahedron that I made from a broken Pyramorphix (2x2x2 shapemod), which acts similar to the Bolaris Ball, but has a different shape (and funny parity case ):

Or perhaps a Missing Link (or Whip It! Puzzle), where you again move the tiles around one by one, but you can also rotate some/each layer(s):

And there are many, many more examples to name. Here in the Moving Holes category of the Twisty Puzzles Museum are quite a few to be found. And some other categories might also be interesting to look into.

• That is an impressive collection! And now I understand your icon too – Beastly Gerbil Oct 27 '16 at 18:59
• I could never have imagined getting so many interesting recommendations; It's sad that I can upvote each answer only once! – errantlinguist Oct 28 '16 at 7:23
• @errant U can offer a bounty :P – ghosts_in_the_code Oct 28 '16 at 11:55

Kevin Cruijssen gives some good examples. You might also want to consider multiple layers as a part of the puzzle, which allows you to have additional constraints, either visible or hidden.

For example, "One Fish, Another Fish", where the frame and piece shapes constrain movement

I highly recommend looking at Rob's Puzzle Page ( http://robspuzzlepage.com ), specifically the page on Rearrangement puzzles. That page lists a huge number of variations, with "3-Dimensional Sliding Piece Puzzles" probably being the most applicable. You'll have to search for that text or scroll past all the twisty puzzles to find it.

Also check out https://sites.google.com/site/geduldspiele/GallerySlidingPuzzles for even more variations on flat sliding puzzles, some of which are multi-layer.

Some specific examples of different types include:

the Pepsi Can puzzle (a cylindrical mapping of a sliding tile puzzle)

the Orbo puzzle (where balls can be pushed in and moved to a nearby empty hole)

the Atomic Chaos puzzle (where you can move balls from one tube to another)

• I could never have imagined getting so many interesting recommendations; It's sad that I can upvote each answer only once! – errantlinguist Oct 28 '16 at 7:21
• @errantlinguist you can offer additional bounty to good answers, if you wish. – Matsmath Oct 28 '16 at 7:25

I'm sure you know of the Rubik's Cube. That has got multiple movable pieces.

If you want something with a lot more movable pieces you could try some of the family members of the Rubik's Cube.

For instance the Gigaminx:

Or even the Teraminx:

There are weirdly shaped ones like the Ghost Cube:

Or Fisher Cube:

And even jointed ones like the Siamese Cube:

So I advise looking at those

• Yes, that is interesting, but the bad part about them is that you move multiple pieces at a time, and I need a game where you only move one at a time. – errantlinguist Oct 27 '16 at 15:15
• Why the Terraminx, and not the Petaminx? Or perhaps Yottaminx? ;) – Kevin Cruijssen Oct 27 '16 at 16:58
• Is the Fisher Cube any different puzzle than the traditional Rubik Cube? It looks exactly the same (mathematically) to me. – The Vee Oct 27 '16 at 20:26
• @TheVee well I haven't got one so I wouldn't know but to start with you can twist diagonally – Beastly Gerbil Oct 27 '16 at 20:27
• @BeastlyGerbil Well, the shape is different but I think the joint in the middle is the same. One just needs to view the top and bottom edge pieces as one normally would the corner pieces and vice versa, and in the middle layer, the role of edge pieces and centre pieces is similarly reversed. – The Vee Oct 27 '16 at 20:30

I like @Beastly-Gerbil's answer, but if you were looking for a 2D puzzle, maybe you could design one with hexagonal pieces, that way any blank spot could have 6 possible pieces moved into its place, which is higher than the 4 in the 15-puzzle and lower than the 8 which you'd get by including corner adjacency.

• That is definitely an improvement, but the problem with diagonal movement isn't due to the complexity but rather that all the moveable pieces are still located near each other (they are all physically adjacent to each other). – errantlinguist Oct 27 '16 at 15:48

Since you've tagged this with sliding-blocks, I assume you're familiar with the sliding block puzzles beyond the fifteen puzzle. The most common one I've seen looks like this:

That doesn't have much more freedom than the fifteen puzzle, but I designed a sliding domino puzzle called Donimoes that has more freedom.

Instead of having the frame to contain the movement, I use rules about matching numbers. Instead of moving the pieces into numerical order, the goal is to slide them into a rectangular shape without matching any numbers.

Here's a simple example:

Visit the web site for the full rules and 20 problems.

You could use a honeycomb layout of hexagonal tiles, which would give you 6 possible moves.

However the geometry means that the tiles couldn't be touching, because they couldn't slide past each other. Instead, they would have to be circular quite a bit smaller than the full hexagon size.

It you are building a virtual game, this wouldn't be a problem - the tiles could just ooze past each other. But if you're building a physical game, you'll have to solve that engineering problem.

• That is indeed an interesting idea, and the actual physics of the board are not important because it'll either be made with cardboard cutouts or done on a computer screen. However, I need to find a way to keep the moveable pieces from being "near" (i.e. adjacent to) each other on the board, so you can't just think e.g. "move the one to the left [of the free space]". – errantlinguist Oct 27 '16 at 15:55
• @errantlinguist what do you mean "keep them from being near"? – Bohemian Oct 27 '16 at 16:03
• Okay, I tried to add a better description in the question... Does that help? – errantlinguist Oct 27 '16 at 16:21

I wrote Knight's Puzzle a while ago. It is based on the 15-puzzle, but only allows Knight moves.

The main difference is that although the position and count of the 2-, 3- and 4- squares (by number of moves available) is the same, they are accessible differently. For example, a 4- square is accessible by 2 3- squares and 2 2- squares.

A 3- square has a 4- and 2*3- as neighbours, and a 2- squares has 2*4- squares as neighbours.

• Interesting, but I have no idea how to solve it in the right way, just try to jump randomly, and make it more scrambled. – Jamal Senjaya Mar 12 '17 at 9:44

LoopOver by CaryKH could be an option. You can go here >>> https://www.openprocessing.org/sketch/580366/

It's basically a sliding puzzle, but as the title suggests, the pieces loop over. Instead of following normal sliding puzzle rules, all the places are filled in, and when you drag a piece, the piece at the end gets pushed into the frame and loops over to the start of the line.

You can also change the size, you can time yourself, do it "blindfolded" and see you averages and best times!

The way the website calculates your averages goes like this: First, it removes your best and worst times. (To make it fair) Then, it takes the average of the remaining times. Sort of how your time is calculated in a Cubing competition.

I hope this works!

• This is also Simon Tatham's Sixteen puzzle. – noedne Mar 27 at 17:19