An Atsumari puzzle begins as a set of numerical clues placed on a game board of hexagonal spaces. The goal of the game is to fill the rest of the hexagonal spaces on the game board with either a white tile or a black tile. On the iPhone/iPod version this means tapping on the game board (see the in-game instructions here). On a printable puzzle this means penciling a dot into an empty hex to indicate a white hex or shading an empty hex to indicate a black hex. There are just three simple rules that determine whether an empty hex should be black or white. The correct solution to an Atsumari puzzle does not violate any of these three rules.
RULE 1: White Cluster Rule
Every one of the numerical clues (a numbered white hexagon) must be a part of a cluster of exactly that many white hexes. A white cluster is enclosed by black hexes or the board boundary. The numbered hex counts (so, a 1-hex is already complete, a 2-hex requires one additional white hex to be attached to it, a 3-hex requires two additional white hexes to be attached to it, etc.). You can only have one numbered white hex per white cluster. Finally, the 0-hex, a unique feature of Atsumari, is also part of the White Cluster Rule. The 0-hex indicates that a white hex is not allowed in that space (i.e. 0 white hexes) and the 0-hex must therefore be black.
RULE 2: Black Cluster Rule
All of the black hexagons must be connected across the game board in one large cluster. This includes the 0-hexes.
RULE 3: Hex Cluster Rule
A big hexagonal cluster made of 7 black tiles is never allowed within the solution. The red and blue dashed hexagons in the diagram above indicate this "dark hex" shape. Another way of stating Rule 3 is that a black hexagonal tile cannot be completely surrounded by six neighboring black tiles. It is important to note that this rule only applies to the black tiles. It is perfectly fine to have a hexagonal cluster of 7 white tiles.
BASIC TIPS and MORE INFO
- The 1-hexes are a big clue for filling up the board and solving the puzzle. Surround 1-hexes with black hexes as you start the puzzle.
- There are only two choices. If you can prove that a hex can't be white then it has to be black.
- Don't join clusters. If an empty space is neighbor to more than one white cluster (see the +'s in the diagram below) then the space must be black. In other words, put black hexes into the gaps between white clusters.
 - Look out for the algebraic puzzles "Solve For X", "Congruence", "Inflation", and "Harmony" in the iPhone App (and there are more on the Challenges page). These puzzles include a symbol 'X' instead of a specific numerical clue. The player needs to find the numerical value of the symbol (e.g. X = 5). Any time a symbol appears in more than one place in a puzzle it has to take on the same value in each place. The correct value for the symbol is the value that makes the puzzle become unique. A solution is considered unique if none of the tiles can be swapped to make a different pattern that still satisfies all of the game rules. Contact feedback@atsumari.com if you need more guidance.
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