Cube 5x5x5

[Julian Gladyshev]

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 Rubik's Professor's Cube is a 5x5x5 variation of the original Rubik's Cube. The first version of this puzzle was invented in 1981 by Udo Krell, who took a regular 3x3x3 Rubik's Cube and expanded its mechanism. Starting in 1983, Krell's design for this puzzle was manufactured and sold by Uwe Mffert. The Rubik's Brand (currently owned by Spin Master) only added this puzzle to their lineup later, in 1986. It is interesting to note that even though this puzzle was not invented by the Rubik's Brand, the name by which it's known today, "Professor's Cube", was coined by Ideal Toys (former owner of the Rubik's Brand) when this puzzle was marketed in the 1980s.

As with all odd-order variations of the Rubik's Cube, the centermost piece on each face is fixed to the mechanism of the cube. Aside from these pieces, there are 48 additional center pieces that are free to move around the puzzle, 8 corner pieces, and 36 edge pieces. Similarly to the Rubik's Revenge Cube (4x4x4), the most common approach for solving this 5x5x5 cube is called the "reduction" method. Focusing on solving the 3x3 center pieces and 3x1 edge pieces first - thus "reducing" the problem to a 3x3x3 puzzle, which can be solved using a regular Rubik's cube solving algorithms. This puzzle has only one parity case.

This puzzle has many moving parts, thus allowing a huge number of possible permutations. The number of possible permutations of the 5x5x5 Professor's Cube is just under 2.83x1074. The 5x5x5 Cube is an official WCA puzzle, and the current world record for solving it is 32.88 seconds. It was set by Max Park from the USA at the CubingUSA Nationals 2023 competition.

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.

For a simpler solution path, I again suggest filling the center cube first to reduce symmetries, but make sure cube 1 or 5 is on the top layer (or bottom, depending on which you want to fill first). Then you can fill the first layer (possibly prioritizing the corners), followed by the subsequent layers.

A Light emitting diode (LED) is essentially a p-n junction diode. When carriers are injected across a forward-biased junction, it emits incoherent light. Most of the commercial LEDs are realized using a highly doped n and a p Junction.

LED cubes rely on an optical phenomenon called persistence of vision or POV. If you flash an LED really fast, the image stays on your retina for a little while after the LED turns off. By flashing each layer of the cube one after another really fast, it gives the illusion of a 3D image, this is also called multiplexing.

The LED matrix we decided to use is a 5 x 5 x 5 monochromatic LED Matrix. This is a total of 125 LEDs. The reason we chose this size, is that it provides the best of overall cube size, construction time & easier programming.

The LED cube is made up from 125 LEDs arranged into 5 layers of 25 LEDs each. The display itself is multiplexed so instead of requiring 125 connections it requires one to each of the five layers and 25 to each LED in a layer making a total of 30. The cube is refreshed by a software interrupt routine with each layer active for 2ms, so the entire cube is refreshed every 10mS (100Hz). This results in a display with no visible flicker.

c80f0f1006