Heyawake Matt Morrow Rules and strategies.

Heyawake Rules  1a) Each “room” must contain exactly the number of black cells as stated. Rooms without numbers can have any amount of black cells  1b) A single line of white cells cannot be in more than 2 rooms.  1c) No 2 black cells can be adjacent to each other  1d) All white cells must connect

Heyawake Contradictions White = Open Black = Closed Gray = Unknown 2a1) There are fewer black cells in a completely filled room than the room number states 2a2) There are more black cells in a completely filled room than the room number states Each “room” must contain exactly the In violation of 1a) Each “room” must contain exactly the number of black cells as stated number of black cells as stated:

Heyawake Contradictions White = Open Black = Closed Gray = Unknown A single line of white cells cannot be in In violation of 1b) A single line of white cells cannot be in more than 2 rooms: 2b) A single line of white cells is in more than 2 rooms

Heyawake Contradictions White = Open Black = Closed Gray = Unknown No 2 black cells can be adjacent to each In violation of 1c) No 2 black cells can be adjacent to eachother: 2 black cells are adjacent to each other 2c) 2 black cells are adjacent to each other

Heyawake Contradictions White = Open Black = Closed Gray = Unknown All white cells must connect: In violation of 1d) All white cells must connect : Some white cells are closed in by black cells 2d) Some white cells are closed in by black cells

Heyawake Derived Rules White = Open Black = Closed Gray = Unknown 3a1) If the number of unknown cells equals the number of remaining black cells, all unknown cells in the room are black. 2 2 : Derived from 2a1):

Heyawake Derived Rules 2 White = Open Black = Closed Gray = Unknown 3a2) If the number of black cells in a room = the desired number, all other squares in the room are white. 2 : Derived from 2a2):

Heyawake Derived Rules White = Open Black = Closed Gray = Unknown 3b) A line of white which originates in a room and then enters another room must encounter a black square before it can enter a third room.1111 : Derived from 2b):

Heyawake Derived Rules White = Open Black = Closed Gray = Unknown 3c) All adjacent squares around a black cell are white. : Derived from 2c):

Heyawake Derived Rules White = Open Black = Closed Gray = Unknown 3d) A white cell which is blocked on 3 sides (by border or black cells) must have its remaining side white. : Derived from 2d):

(2x-1)-by-1 Room Rule 3 White = Open Black = Closed Gray = Unknown An area which must contain x black cells and whose dimensions are 1 by (2x-1) can only have one configuration: black cells at the ends, and then alternating black and white cells. This can be derived from repeated 3a1 and 3c.3

Inside x-by-2 Room Rule 3 White = Open Black = Closed Gray = Unknown An area which must contain x black cells and whose dimensions are 2 by x has exactly 2 configurations (3a1, 3c)3 3 Since no cell can be adjacent to another, there can only be 1 cell per row and they must zigzag.

Outside x-by-2 Room Rule 3 White = Open Black = Closed Gray = Unknown We can use a 2 by x area containing x for a proof by cases since we know the configuration can either be one way or the other. (uses 3d and Inside x-by-2)33 3

3-by-2 Border Room Rule 3 White = Open Black = Closed Gray = Unknown Red = Wall A 3-by-2 room next to the border has only one solution:3 3 Contradiction! (2d) Use Inside x- by-2 rule for cases

Heyawake Room Cases White = Open Black = Closed Gray = Unknown Red = Wall In a 3 by 3 area with 4 remaining, if a black cell is a middle edge cell, all the black cells must be middle edge cells (3d, Inside 3-by-2). This results in a single white cell completely surrounded by black. ┴ (2d) therefore none of the middle edge cells can be black and must be white. 4 5 The only way in which 5 black squares can fit in a 3 by 3 is in the following configuration:

LEGUP Issue  Heyawake is region based  Need a class which would be like the boardstate class where this could be loaded and checked  Not necessary for the regions to change  Is this reasonable to assume for all puzzles? (killer sudoku)  Regions need to have a draw outline function  How would we represent a region?