Solve 5x5x5 Cube
Yufei Labbe
The figures in this document represent a sample cube. Although the colors on your personal cube may be different than the figures, I feel that the colored figures will still be helpful in solving the cube. I would suggest matching as many colors on your cube as possible to the figures, then making mental notes about which colors correspond between your cube The diagrams show all six sides of the cube by pretending that mirrors are being held up so that you may see the "hidden" sides.
The directions for what parts of the cube to turn and when are given in a code that is relative to the current positioning of the cube. Each side descriptor refers to a side with respect to the figure the move is referenced to. This means that the front side can be a different color, depending on the diagram being used. See Move 1 of the 3x3x3 Rubik's Cube page for instructive diagrams to illustrate the moves.
Since the lower case L: l looks like the number 1, a letter by itself means to turn the side clockwise one quarter turn. A letter with a 2 following it means to turn the side two turns (halfway around). A letter with an apostrophe (') means to turn the side counterclockwise one quarter turn (a -1 can be used in place of an apostrophe, but the apostrophe takes up less space).
In the figures, any gray pieces represent pieces that don't matter and can be any color. To increase clarity, I have shown a lot of colored pieces. In most cases, the figures represent an example of the situation being described, while the colors don't have to match perfectly.
Pick a center piece to represent the first side of the cube solved (blue in these figures) and solve the points and crosses on that side. One cross may be left empty since it will be used as the "keyhole" later on.
Orient the cube as shown in Figure 2a, where the two points to be moved into place are on the right side of the front face while any green points already on the down face are on the right side of the down face, then perform Move 2a.
In words, rotate the Right side and 2nd layer in from the right side counterclockwise one turn, the down (bottom)side two turns (180 degrees) and the right side and 2nd layer in from the right side clockwise one quarter turn.
Orient the cube as shown in Figure 2b, where the one point to be moved into place is on the right side of the front face while any green points already on the down face are on the right side of the down face, then perform Move 2b.
Use your favorite 3x3x3 solution to solve the cube, since the edges and wings act like the single edge pieces of the 3x3x3, the corners directly correspond to the corners, and the centers, crosses, and points correspond to the center pieces of the 3x3x3, as shown in Figure 7.
You guys have repeatedly requested that we make a Rubik's Cube 5x5x5 Solver so we decided to give it a try. We are proud to present the world's FIRST and BEST online Professor's Cube Solver. This is still an early version so we will appreciate your feedback.
The Rubik's Professor's Cube (5x5x5) has about 283 trevigintillion different possible combinations. We know you've never heard of "trevigintillion" but trust us it's a LOT - way more than the original Rubik's Cube's 43 quintillion possible combinations.
Like our Rubik's Revenge Solver (4x4x4), this solver was programmed to use the "reduction method" - meaning it will solve the centers and edge pieces first, then solve the rest of the puzzle as if it was a normal Rubik's Cube (3x3x3). Note that this is by no means an optimal solver and will take around 100 moves to solve a random combination. We know it's a lot but if you want your puzzle solved you'll have to put in the time to paint the 3D model and follow the step by step solving instructions.
Use the color palette to paint the cube - select a color by clicking or tapping it, then click or tap the tiles you want to use the selected color for. Drag or swipe the cube to rotate it. When finished hit the "Solve" button and the step by step guide for solving your Professor's Cube 5x5x5 will be displayed to you.
The 5x5x5 Rubik's Cube also known as the Professor's Cube is the invention of Udo Krell. On this page I'm going to present this puzzle and will give you a clue about how to solve the 5 layered Rubik's Cube.
Centres are solved similarly to a 4x4x4, in which 'strips' of inner pieces are created and lined up in place. First create the centre strip around the centre piece (conventionally around the white centre), then create more and move them into place. Continue this until there are two adjacent centres to complete, in which case in like to treat as if I'm solving one of them, in which the other ends up solved.
I have this wooden puzzle composed of 25 y-shaped pentominoes, where the objective is to assemble them into a 5x5x5 cube. After spending quite a few hours unsuccessfully trying to solve the puzzle, I finally gave up and wrote a program to try all possible combinations using backtracking. Analyzing the results revealed that for every solution found, the computer made - on average - 50 million placements and removals of pieces. This is obviously beyond my capabilities as a human, even if I can see a few steps ahead that a partial solution leads to a "dead end".
Get Burr Tools. It takes less than a minute to set up this problem, and then a few minutes to generate 1264 solutions. I'm not sure if that's all solutions, the solver tells me 22 minutes are now needed to completely check the solution space. (EDIT -- Total solving time = 24.8 minutes)
First, a few thoughts about this specific box packing problem. It might be possible to pack this box by packing the "y" polycube into 25 cells in 5 identical layers. This corresponds to collapsing this 3-D shape into a polyomino in the plane, and then using it to pack a 5x5 rectangle. However, this polyomino can not tile a 5x5 rectangle. (The terminology for polyominoes is not totally standard but some use the order of a polyomino to mean the smallest number of copies of this shape to tile a rectangle with congruent copies of itself.) The order of this polyomino is 10.
Fourth, there are "complexity" results which govern the general problem of given a collection of tiles (either in 2-D or 3-D) when can one use them to tile some particular shape (say a rectangle or rectangular box). Here is a paper related to this of Erik Demaine:
While you talk about pruning the search tree through symmetry, generally the best way I've seen to approach these problems is by breaking cells down into classes or layers; for instance, many of the classic polyomino non-tiling problems can be solved with suitable colorings, and constraints provided by colorings can help reduce the search space immensely (e.g., the Soma cube result that the 'T' piece must have its spine on an edge of the cube). The thing that jumps out at first glance is the 'intermediate' layer of cells, the 26 cells that are neither on the outermost face of the cube nor the center cell. Whatever piece occupies the center must also occupy at least (and in fact, exactly) three of these cells, and any piece that's not wholly on an outer layer occupies at least one, with many placements forcing several cells to be occupied; it seems intuitively like they might be at a premium and that that's a constraint worth investigating. The section on the Soma cube in the last volume of Winning Ways For Your Mathematical Plays has some discussion of these sorts of colorings, and that might be a good place to start.
I realized about 80% through that this answer may have too many parts to implement, so I'm also going to leave a shorter one.
If you prioritize filling some cells over others, you can at least prune the computer's search tree. I suggest filling the center cube, followed by the six face centers, followed by the eight corners, finishing with everything else.
Here, let me number the squares of a Y pentomino:
1
23
4
5
By starting with the Y filling the center cube, the search space can be reduced by a factor of 12 by holding this piece's orientation constant. Note that only cells 2, 3, and 4 can occupy the center of the cube. If 2 or 4 is the center cell, then five face centers will remain to be filled; if it's 3, then all six will remain.
For the five or six centers, the Y occupying each center can be placed in one of 36 ways, disregarding overlap, as follows:
-The "spine" (the cubes 1245) can point in any of four directions
-Again, only 2, 3, or 4 can fill these cubes
-The full Y can lie on the face in three ways for each cube: on either Y face for all three, plus either along the spine for 2 and 4 or alongthe furthest opposing face for 3
Now for the eight corners. Only 1 and 5 can occupy these cubes, so for each of these corners we can work on partial placements for just the spines along the edges. Now, each corner is adjacent to three three-square edges, although some of the twelve edges are certainly part-occupied by Y's placed in the previous phase. The corners to be filled need to be prioritized by fewest available edges.
-If, at any point, an unfilled corner has 0 free edges, then either the last "temporary edge placement" or the entire face-center configuration needs to be advanced to the next one
-Whenever a corner piece is laid along an edge and the other corner sharing that edge is not yet filled, that other corner should be re-prioritized for having one fewer free edge
-If some unfilled corner has only one free edge, a spine is placed along that corner and edge
-And if no unfilled corner has fewer than two free edges, pick and note a "temporary edge placement" to fix if it leads to a contradiction. (I guarantee at least two corners will have no more than two edges. If the first placement is wrong, the second placement should be used but not noted as temporary.)
This gives eight partial placements and six or seven total placements. This leaves eleven or ten Y's to place within a volume of 63 or 58, respectively, which includes at least 26 cubes that are absolutely not filled regardless of how exactly the corner Y's are placed (each one still has up to four possibilities for the placement of the 3 cube), a much smaller search space.