Rätsel und Puzzles

# Rules

The different rule descriptions use common phrases which are explained in detail on this page.

### Neighborship:

A cell (red) has 3, 5 or 8 neighbors (blue and green).

The blue cells are the orthogonal (horizontal and vertical) neighbors of the red cells.

The green cells are the diagonal neighbors of the red cells

### Regions, Areas, Stripes:

A region consists of one or more cells which are separated by bold lines from other regions.

An area consists of one or more cells which are not separated by bold lines from other areas, but defined otherwise, e.g. all adjacent cells of the same color. In the diagram the yellow cells form such an area.

A stripe is a region (green) or an area (red) which is exactly 1 cell wide and of any length.

Areas of black cells are sometimes called blocks or groups, or dominoes, triominoes, tetrominoes, pentominoes, or polyominoes.

Regions as well as areas can be square, rectangular or irregular. (Note: A square is also a rectangle!)

### Snakes

A "snake" is a path through the cells of the diagram which is exactly one cell wide. In many genres a snake must not touch itself, not even diagonally.

A snake may be open (i.e. the snake has a head and a tail) or closed (i.e. a loop)

### Orthogonally contiguous cells/areas

The white cells form an orthogonally contiguous area, if it is possible to travel from any white cell to any other white cell, moving only horizontally and vertically (never diagonally) and never crossing a black cell.

In the diagram the white cells are colored blue, green and yellow (actually, they all are white).

• All green cells are orthogonally contiguous: For example, one can travel from A to B moving only horizontally and vertically, not crossing a black cell.

• Also, all blue cells are orthogonally contiguous.

• The green cells and the blue cells are not orthogonally contiguous. For example, there is no path from C to D, travelling only horizontally and vertically which does not cross a black cell.

• Cell E is isolated and is not orthogonally adjacent to any other white cell.

Orthogonally contiguous areas of black cells are defined analogously.

### Dominos

A domino is a region of exactly two orthogonally adjacent cells. Disregarding rotation and reflection there is exactly one domino:

In the diagram you can find 3 Dominos. Yellow and blue have the same orientation; yellow and green have different orientation. Yellow and green are orthogonally adjacent; yellow and blue are diagonally adjacent.

### Triominoes

A triomino is a region of exactly three orthogonally adjacent cells. Disregarding rotation and reflection there are exactly two triominoes, labeled by letters according to their shapes:

In the diagram you can find 3 triominoes. The orange and the blue triomino together cover a 2x2 area which is not allowed in many puzzle types.

### Tetrominoes

A tetromino is a region of exactly four orthogonally adjacent cells. Disregarding rotation and reflection there are exactly five tetrominoes, labeled by letters according to their shapes:

In the diagram you can find 3 Tetrominos.

### Pentominos

A pentomino is a region of exactly five orthogonally adjacent cells. Disregarding rotation and reflection there are exactly twelve pentominoes, labeled by letters according to their shapes:

### Polyominoes (N-ominoes)

A polyomino is a region of orthogonally adjacent cells; an N-omino is a region of exactly N orthogonally adjacent cells. Dominoes, triominoes, tetrominoes and pentominoes are special cases of N-ominoes (N=2, 3, 4 or 5).

### Congruence

Two figures (areas, domains, polyominos) are congruent if they can be made to coincide by translation, rotation, mirroring, or a combination of these methods. [Wikipedia]

The following pentominos are all congruent:

### Manhattan Distance (Taxi Distance)

The Manhattan Distance is the shortest link between two cells, travelling only horizontally and vertically between adjacent cells. The diagram shows three possible paths; all paths are equally in length.

Formally: The Manhattan Distance between two cells is the sum of the absolute differences of the coordinates of the cells the path travels through:

In the diagram the Manhattan Distance between the two blue cells is 7 (= number of cells the path goes through including the target cell).