2
votes
1answer
277 views

What is the probablity that a Rubik's cube is solvable if you randomly switch two squares?

I am trying to learn how to solve a Rubik's cube and the 4x4 rubic cube at school is missing 4 squares, that got me thinking: if you randomly switched two squares, would it still be solvable? I ...
8
votes
3answers
192 views

Group of sphere transformations, impressing friends

Ok, so here's the story: I am reading a book on algebra and, via some exercises, discovered that in any group $G$, the order of $x \cdot y$, written $o(x \cdot y)$, equals $o(y \cdot x)$. Now, this is ...
7
votes
1answer
148 views

“Multi-facets” rope puzzle

I've done the following, can you tell me if it's correct? If $n$ is the number of sides of the rope and $k$ is the number of rotation, e.g. $k=0$ for glue each side to itself then I think the ...
5
votes
4answers
943 views

Calculating Gröbner basis for Sudoku

I'm trying to write a program that solves sudokus using a Gröbner basis. I introduced 81 variables $x_1$ to $x_{81}$, this is a linearisation of the sudoku board. The space of valid sudokus is ...
14
votes
1answer
298 views

How many strategies are there for this puzzle where one of n logicians must call his own hat's color among n?

$n$ logicians are wearing hats which can be of $n$ different colors. Each logician can see the colors of all hats except his own. The logicians must simultaneously call out a color; they win if at ...
5
votes
3answers
170 views

What are good ways of understandng a permutation group from a set of generators?

I'm trying to understand the structure of a Rubik's Cube-style puzzle I'm playing with; I have an expression of the solutions as the permutation group generated by four elements of $S_{16}$, each a ...