Rubik 39;s Cube 5x5x5 Solver
Lorriane Nasuti
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.
Cube Explorer implements an algorithm which is capable of doing this and it finds the solution usually within fractions of a second.
The program also gives the possibility to generate cubes with certain patterns or symmetries.
The Fifteen Puzzle Optimal Solver solves any given position in the minimal number of moves. Though the complexity of this puzzle is lower than that of Rubik's Cube, the task of finding an optimal solution for this puzzle is not always easy. If you find a solver or you have written a solver which runs considerably faster than the one provided here please let me know.
We present an algorithm which optimally solves this kind of puzzles quite effectively and provide a program which implements this algorithm and which you can use to create, analyze and solve puzzles of this kind.
The word "kaleidocycle" was coined by Wallace Walker in the 1950s. Together with Doris Schattschneider he published a book in 1977 [1] using patterns of M.C. Escher which contributed greatly to the spread of this word.
We take a closer look at kaleidocycles which are made out of 6 disphenoids (isosceles tetrahedra) and answer the question for which triangle side lengths a, b and c such kaleidocycles are physically realizable such that the parts do not block each other during the movement.
Paul Schatz was a German mathematician and inventor with a fable for anthroposophy. One of his discoveries was the inversion of the cube. Stimulated by Paul Schatz's work, several people (among them Klaus Ernhofer, Wolfgang Maas, Immo and Friedeman Sykora) have developed additional inversions of Platonic soldis.
If you cut a dodecahedron into 6 congruent pieces and hinge the parts together as shown in the animated gif above, the forced movement shown there is theoretically impossible because at some points of the movement adjacent pieces slightly penetrate each other.
To remove this shortcoming I computed a slightly irregular dodecahedron which still has threefold dihedral symmetry D3d and which differs from the regular dodecaedron by only about 1% - 2% concerning side lengths and face angles. With its hinged six congruent pieces the inversion process shown is perfectly possible. See here for details.
The Pentarot 2D Rotational puzzle was inspired by my interest in Penrose tilings. It has 36 pentagonal tiles which are scrambled by rotating the 5 rings with 10 pentagons each. Sadly I did not find a way to realize it physically, but you can downlad a Windows program to play with it.
That Penrose tilings and the centuries-old tradition of islamic ornamental designs with five- and tenfold symmetric elements have similarities was noticed by Peter Lu from Harvard University in 2007 and provoked major media interest. It is not clear yet, however, to what extent the ancient architects were aware of the underlying mathematical laws of a Penrose tiling. More recent works by Peter R. Cromwell 2015 and 2016 tend to give a negative answer to this question.
Reflections on these tilings result in a set of tiles in the style of Islamic art which forces a tiling equivalent to a Penrose tiling (or in the exact mathematical terminology: the tilings are mutually locally derivable).
Though there are standard Sudokus which are almost impossible to solve only with logical reasoning by humans, from a computational point of view solving a standard Sudoku is almost trivial. Since the general problem is NP-complete finding a solution gets more demanding for larger grids.
We choose an approach where we simplify the given NxM-Sudoku as far as possible using "human" methods like hidden and naked singles and tuples, block-line interaction etc. and transform the remaining problem into a boolean satisfiability problem. We then use Sat4J for solving.
Max Park is an American Rubik's Cube speedsolver who formerly held the world record average of five 3x3x3 solves (by WCA traditions), 6.39 seconds, set on 23 April 2017 at OCSEF Open 2017.[1] He also holds the world record for the fastest 3x3x3 Rubik's cube single solve, which he set in July of 2023. He finished the solve with a time of 3.13 seconds. Prior to this, the record had been held by Feliks Zemdegs of Australia, who had improved it 9 times over 7 years from 9.21 seconds on 30 January 2010 to 6.45 seconds.[2] Park is the only cuber other than Zemdegs to have set the record since 27 September 2009.[2]
Park holds the world record for average of five 4x4x4 solves: 21.11 seconds, set at Bay Area Speedcubin' 21 2019. He used to hold the world record for a single solve of 18.42, before German speedcuber Sebastian Weyer took it in September of 2019.[3]
Park holds the world records for single and average of five 5x5x5 solves: 34.94 seconds and 39.65 seconds, set at Houston Winter 2020 and CubingUSA Western Championship 2019 respectively.[4] Prior to Park's first 5x5x5 record, the records for single and average of five 5x5x5 solves had been held by Feliks Zemdegs of Australia, who had improved the two records a combined 32 times.[5] Park is the only cuber other than Zemdegs to have set either 5x5x5 record since 11 August 2012.[5]
Park holds the world record for average of five 3x3x3 solves with one hand: 9.42 seconds, set on 16 September 2018 at Berkeley Summer 2018.[8] Park was the first person to achieve a sub-10 second one-handed average in competition, with an average of 9.99 seconds on 13 January 2018 at Thanks Four The Invite 2018. Park also holds the world record single for one-handed solving at 6.82 seconds set at Bay Area Speedcubin' 20 2019, breaking the longest standing cubing world record which was held by Feliks Zemdegs.[9]
Park is one of two cubers to have solved the 3x3x3 in less than 5 seconds in competition at least five times,[12] and one of the two cubers to have achieved at least five sub-6 second averages of five 3x3x3 solves in competition.[13]
One of my hobbies is building Lego Mindstorms robots that can solve rubiks cubes. I was able to find solvers for 2x2x2, 3x3x3, 4x4x4 and 5x5x5 but I couldn't find a solver for anything larger than that :( The solvers that I did find for 4x4x4 and 5x5x5 took quite a bit of RAM (several gigs) but I wanted to be able to run the solver on a Lego Mindstorms EV3 which is 300Mhz and 64M of RAM. So I decided to write my own solver and here we are :)
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.
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