Heyawake

Like most other logic puzzles from Nikoli, the size of the playing field in Heyawake is not fixed. It can be small - 6x6 cells, or large - from 10x10 cells.

The larger the field, the more difficult the puzzle to solve, which makes it easier to rank these games from easiest to hardest. Thus, even a beginner can cope with a 6x6 puzzle, but solving a 20x20 puzzle will require complex mental work and extreme care.

Basic rules

Regardless of the size of the playing field, it is divided into separate zones by lines. In the original version of Heyawake, each zone contained a number indicating how many cells in it needed to be painted over. Today this rule has been abolished, since a solution can be found with partial numbering of zones. So, some of them contain a number, and some do not. Obviously, you need to start the game from the beginning, using the elimination method.

The player has the opportunity to mark not only filled, but also empty cells. If the first ones are completely painted over, then the second ones can be marked with dots, crosses or any other marks (except for full shading).

This makes solving the puzzle easier, since as the game progresses you will have to constantly eliminate obviously incorrect moves, using the method of deduction. In this case, you need to adhere to five basic rules:

• All white (unshaded) cells must be combined into a common network in such a way that from any part of it you can get to any other.
• Shaded cells should not touch each other vertically or horizontally. Diagonal contacts are allowed.
• The number inside the circled area indicates how many shaded cells there should be.
• If there is no number inside the region, it can contain any number of shaded cells.
• At the end of the game, there should be no straight white lines on the field passing through more than two adjacent zones.

Given these rules, Heyawake can be classified as a mathematical problem that has an NP-complete solution. That is, this problem is as difficult to solve as the Boolean satisfiability problem. It is even more difficult to create such a puzzle, so without the use of digital technology the number of games was greatly limited.

How to solve the puzzle

The Heyawake solution can be simplified by the elimination method, based on the five basic rules described above. Once you understand them, you will be able to successfully use deduction during the game.

• If one of the cells is exactly shaded, all cells orthogonal to it automatically remain unshaded and can be marked with a cross.
• If, when painting a cell, it divides the array of white (unshaded) cells into two parts, it can be marked with a cross, since such a move is prohibited by the rules.
• If there is a circled 2x2 zone with the number 2 in the corner of the playing field, it should have one filled cell in the corner of the grid and one cell located diagonally to it.
• In the circled 2×3 zone, adjacent with its wide side to the border of the field and containing the number 2, the cell in the center of the side (with 3 cells) running along the border should be filled in, and 2 cells in the opposite corners of the zone.
• In the 1x3 zone with the number 2, the outer cells are always painted over, and the central one remains unpainted.
• A 3×3 zone with the number 5 is painted over with a mosaic pattern: filling the central cell and 4 cells diagonally.

It is easier to correctly fill the circled area, the smaller its size and the larger the number indicated in it. Therefore, it is always worth starting the game from areas where there are the fewest alternative moves. And only then, using the method of elimination, fill the neighboring playing areas.

Inexperienced players should start with the simplest variations of Heyawake - 6x6 and 8x8. When the winning percentage approaches 90-100, you can move on to larger playing fields - 10x10 or more.