Color Fill (Kami)

Color Fill, popularized by the mobile game Kami, is a puzzle where the player must make the entire board a single solid color within a limited number of moves. The board is divided into irregularly shaped regions, each filled with a color. On each move, the player selects a region and changes it to a new color, which also merges it with any adjacent regions of that same color. The challenge is to find the correct sequence of color changes that unifies the board within the move limit — a constraint that often allows only one solution path.

Rules

1. The board consists of irregularly shaped regions, each a solid color.
2. The player selects a region and picks a new color for it.
3. The selected region changes to the new color.
4. If adjacent regions now share the same color, they merge into one larger region.
5. The goal is to make the entire board one color.
6. Each puzzle has a move limit (a "perfect" score). The player must reach the goal within that limit.

Strategies

• Work from Edges Inward: Start by unifying small regions at the edges and corners, as these have fewer neighbors and fewer options.
• Identify the Final Color: Look at which color is most prevalent on the board. The last move will likely merge everything into this dominant color.
• Chain Merges: Plan color changes that cause cascading merges. Changing one region to a color shared by multiple neighbors creates a large unified area in a single move.
• Bottleneck Analysis: Identify regions that connect two large areas of the same color. Changing the bottleneck region to that color can merge both large areas at once.
• Work Backward: Start from the goal (one color) and think about what the second-to-last state must look like, then the third-to-last, and so on. This often reveals the solution path more clearly than forward thinking.
• Minimize Isolated Colors: Each distinct color remaining on the board requires at least one more move. Prioritize eliminating colors that appear in only one or two regions.

The color fill puzzle mechanic has roots in the broader family of flood-fill puzzles that emerged in the 2000s. The concept of unifying a divided surface by changing colors of regions gained popularity through various web and mobile games.

The most acclaimed version is Kami, developed by State of Play Games and released in 2013 for iOS. "Kami" means "paper" in Japanese, and the game featured a stunning origami-inspired aesthetic where colored paper regions would fold and unfold as the player changed colors. The game's visual beauty, combined with its increasingly challenging puzzles, earned it widespread praise and multiple design awards. Kami 2, released in 2017, expanded the concept with user-created puzzles and a daily challenge mode.

The puzzle type is closely related to map coloring problems in mathematics, specifically the inverse problem: instead of coloring a map with minimum colors so no adjacent regions share a color, the player works to make all regions the same color using minimal moves. This connects to graph theory, where the regions and their adjacencies form a graph, and the puzzle asks for a minimum-length sequence of operations that reduces the graph's chromatic partition to one.

Color Fill puzzles vary in difficulty based on the number of regions, colors, and the topology of the board. Simpler versions use regular grids, while more complex ones (like Kami) use irregular regions with intricate adjacency relationships. The puzzle type has been widely adopted in puzzle game compilations and brain training apps, valued for its blend of spatial reasoning and sequential planning.