“Intelligent Toys” is a puzzle firm within the Czech Republic. Their Website describes 5 totally different hand-made, picket puzzles which can be associated mathematically to the Rubik’s Dice.
Contents
Puzzle
Right here is the photograph of the puzzle “Trio”.
Right here is an English translation of their description.
The duty of this 2D puzzle is to assemble all of the ovals in accordance
to their shade right into a fundamental meeting. By shifting the balls within the
grooves and turning the middle wheel, you possibly can step by step get every
ball the place you want it.
I’m not certain that Intelligent Toys remains to be in enterprise. They haven’t responded to my emails and I’ve not been capable of buy an precise Trio puzzle. After all, that’s all of the motivation I must make a MATLAB mannequin.
Mannequin
Right here is the preliminary configuration. There are three fastened outer, partial, discs and one internal, full, disc. Every of the outer discs has a channel containing 10 marbles. When the internal disc is positioned correctly, the marbles in a channel will be rotated. Rotating the internal disc itself strikes some, however not all, of the marbles, thereby scrambling the colours.
Like Rubik’s Dice, the target of the puzzle is to return to this preliminary “solved” state.
Mathematically, each Rubik’s Dice and Trio are finally primarily based upon rotation matrices. Rubik’s Dice is a 3-D puzzle whose state is specified by the place and orientation of 27 cubelets; there are six colours and, at every step, 12 doable rotations. This provides Rubik’s Dice about 4.3*10^19 doable positions.
Trio is a 2-D puzzle whose state is specified by the colours of 10 marbles in every of three channels; there are 10 marbles of every of three colours and, at every step, eight doable rotations (4 discs, clockwise or counter-clockwise.) This suggests that Trio has fewer than 3^30 = 2.1*10^14 doable positions.
Scramble
A “scramble” is an integer vector with parts between -4 and +4 that specify strikes or rotations. Transfer 0 initializes the mannequin. Transfer d with d equal to 1, 2, or 3 rotates all of the marbles within the d-th channel counter-clockwise for one-tenth of a full rotation. Transfer d with d equal to -1, -2, or -3 is the corresponding clockwise rotation. Strikes -4 and 4 flip the central disc clockwise or counter-clockwise for one-third of a full rotation. This rotates some, however not all, of the marbles and mixes the colours.
Here’s a scramble of size 29 that gives our instance.
D = [ 4 1 3 1 1 -2 3 1 4 4 2 4 4 -3 -3 -2 -3 -3 2 1 4 -1 3 -4 2]
And right here is the scrambled outcome.
Animation
This animated gif reveals the scrambling course of one transfer at a time. The animation doesn’t repeat routinely, so to start out it over once more, refresh your browser. If it nonetheless would not transfer, discover one other browser.
Unscramble
Reverse the scramble by working it backwards, altering the signal of every transfer. This may return the scrambled place to the preliminary place. I name this “unscramble”; it solves the scrambled place by a “observe the breadcrumbs” algorithm.
Each animations take a very long time to run — about 80 seconds with my browser.
Options
I haven’t got any thought about tips on how to really clear up a given place with out utilizing information of the way it was generated, and I haven’t got any thought about tips on how to quantify the issue of discovering an answer. That is in sharp distinction to Rubik’s Dice the place there are measures of issue and algorithms for locating optimum options.
Software program
This system obtainable at this hyperlink is interactive. Click on or alt-click in any one of many 4 discs to make a transfer in that disc.
Revealed with MATLAB® R2022a
