Linked Questions

23
votes
1answer
4k views

How many ways are there to fill a 3 × 3 grid with 0s and 1s?

Extra conditions that put a formal solution out of my reach: the centre cell must contain a $0$, and two grids are equal if they have a symmetry, e.g. $$\left( \begin{array}{ccc} 0 & 1 & 0 \\ ...
16
votes
2answers
1k views

How many 2-edge-colourings of $K_n$ are there?

I'm writing a paper on Ramsey Theory and it would be interesting and useful to know the number of essentially different 2-edge-colourings of $K_n$ there are. By that I mean the number of essentially ...
9
votes
3answers
2k views

Cube stack problem

How many distinct patterns are possible if you omit (a) 1 piece, (b) 2 pieces and (c) 3 pieces from a cube originally consisting of 27 smaller and equally sized cubes?
2
votes
2answers
1k views

Counting Shaded Squares

In a $4 \times 4$ square, how many different patterns can be made by shading exactly two of the sixteen squares? Patterns that can be matched by flips and/or turns are not considered different. How ...
3
votes
1answer
2k views

In how many different ways can the faces of a regular dodecahedron be colored?

prove that there are 9099 different ways of colouring the faces of a dodecahedron red, white or blue. (this is from Amstrong's "group and symmetry") attempt: Since the question refers to the faces of ...
0
votes
2answers
2k views

coloring 3x3 chessboard with two colors…

Consider a 3x3 chessboard with 9 elements. The elements are colored with black and white paints. The task is to find the number of different chessboards of this type exist. I don't know how to ...
3
votes
3answers
401 views

TicTacToe with considerations of symmetry

It is not difficult to determine the number of possible games of tic toe, but what about when rotationally symmetric positions are considered equal? Please do not simply give me the number, I would ...
3
votes
1answer
740 views

A 4 × 4 grid of squares is filled in, with each of the 16 squares colored either black or white…

A 4 × 4 grid is filled in, with each of the 16 squares colored either black or white. Two colorings are regarded as identical if one can be converted to each other by performing any combination of ...
4
votes
1answer
133 views

Amount of 3 x 3 block of binary pixels combinations with rotations/reflections considered identical

I am considering a $3\times 3$ block of pixels where each pixel can be either $0$ or $1$. How many unique blocks are there if we consider rotations/reflections to be identical? I believe the answer ...
1
vote
0answers
140 views

Given N blocks, find the number of unique shapes in a NxN block

Constraints: The blocks must be adjacent to each other. i.e. A pair of blocks must have a common edge or vertex. Any shapes that are formed by flipping or rotating or mirroring should be considered to ...