Solution to n-dimensional Ruby-cube.
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Not sure how much perfect / imperfect is this evaluation.
Let us have an n-dimensional Ruby-cube with all six faces having well-arranged so that any of the two squares on a face are of the same color.
Now let us rotate one or more faces m times in whatever random way but note-down the direction (clock-wise or anti-clockwise) of every movement correctly.
Thus we get a particular though jumbled-up set-up of the squares on each of the six faces.
Now if we rotate these faces in the reverse direction and exactly opposite to the one we had rotated before having got this set up, we shall come upon the initial position.
Now, as in this way we can reach all possible jumbled-up set-up.
(Hint : though a very large number indeed, but is finite and could be worked out by calculating all possible 'permutations and combinations' using factorials).
We see that each of them has a unique solution.
As we can mathematically find out this number how-so-ever big, each has this unique solution with minimum number of moves which takes us to the initial position.
--
A friend suggested there may be several solutions with lesser number of moves.
Of course!
Here I wanted to point out there is at least one possible solution for each and every jumbled-up set of squares. Though the one with minimum number of moves may be yet another, unique....
Hope Manjul Bhargava likes this.
--
--
Not sure how much perfect / imperfect is this evaluation.
Let us have an n-dimensional Ruby-cube with all six faces having well-arranged so that any of the two squares on a face are of the same color.
Now let us rotate one or more faces m times in whatever random way but note-down the direction (clock-wise or anti-clockwise) of every movement correctly.
Thus we get a particular though jumbled-up set-up of the squares on each of the six faces.
Now if we rotate these faces in the reverse direction and exactly opposite to the one we had rotated before having got this set up, we shall come upon the initial position.
Now, as in this way we can reach all possible jumbled-up set-up.
(Hint : though a very large number indeed, but is finite and could be worked out by calculating all possible 'permutations and combinations' using factorials).
We see that each of them has a unique solution.
As we can mathematically find out this number how-so-ever big, each has this unique solution with minimum number of moves which takes us to the initial position.
--
A friend suggested there may be several solutions with lesser number of moves.
Of course!
Here I wanted to point out there is at least one possible solution for each and every jumbled-up set of squares. Though the one with minimum number of moves may be yet another, unique....
Hope Manjul Bhargava likes this.
--
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