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Herbert Kociemba's Homepage


 Rubik's Cube and Cube Explorer
 15-Puzzle Optimal Solver
Kaleidocycles with 6 Disphenoids
  Inversion of a Dodecahedron
  Pentarot 2D Rotational Puzzle
 General NxM-Sudoku solver and generator
  The 125 cube: Packing a 5x5x5 cube with Y-pentominoes
 Penrose Tilings and Islamic Ornamental Design
 The Spiral of Theodorus

Restoring a Rubik's cube is not the easiest task but if you want to do this in 20 moves or less the bar has been raised even higher.

 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 15-Puzzle is a sliding puzzle that triggered a similar hype in 1880 as Rubik's cube did exactly 100 years later.

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.

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.

A Penrose tiling consists of a set of tiles which tesselate the plane with no overlaps and no gaps. The tiling was invented 1973 by Roger Penrose.

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).

The NxM-Sudoku is a generalization of the standard 3x3 Sudoku with 9 squares of size 3x3 and a total of 81 cells. In the general case we have N*M rectangles of size NxM and a total of (N*M)2cells.

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.

Besides solving the program also includes features like generating Sudokus, testing for unique solutions etc.

The 125puzzle is a quite difficult puzzle where you have to place 25 Y-pentacubes in a 5x5x5 box. I never managed to solve the puzzle manually.

But I successfully could solve the puzzle using a SAT-solver and present 5 solutions with a special property.

As a byproduct I also give solutions for 20 Y-pentacubes in a 5x5x4 box, for 16 Y-pentacubes in a 4x4x5 box, for 12 Y-pentacubes in a 4x5x3 box and for 12 Y-pentacubes in 5x6x2 box.

The Spiral of Theodorus, eventually vaguely familiar from school mathematics provides certainly the opportunity for investigations which goes beyond the subject matter at school.

  On this page wie find with the help of the Euler–Maclaurin formula arbitrary good approximations for the discrete and analytic version of the square root spiral. We compute the so called Schneckenkonstante K to more than 500 decimal places and visualize the used methods in this Geogebra-Applet.

© 2020  Herbert Kociemba