Rules
''Fillomino'' is played on a rectangular grid with no standard size; the internal grid lines are often dotted. (When published as ''Allied Occupation'' in theSolution methods
It is common practice in solving a ''Fillomino'' puzzle to add numbers to the empty cells when it is determined what size polyomino each must belong to; these numbers are effectively treated identically to the givens. As well as making it clear where many border segments must be drawn - such as between any two differing numbers, or surrounding a region of matching numbers whose quantity is that number - it also permits the second part of the puzzle's rule to be visualized as simply "the same number cannot appear on both sides of a border", which greatly accelerates solving. A curious side effect of numbering every cell is that when the puzzle is completed, the numbers alone unambiguously define the solution, the actual borders being trivially deducible. This makes communication of a solution without a grid quite feasible; indeed, solutions for ''Allied Occupation'' give only the numbers. ( Nikoli always publishes solutions to their ''Fillomino'' puzzles with both the polyomino borders drawn in and numbers given in every cell.) The typical means of starting a ''Fillomino'' puzzle is to draw in the obvious borders between non-matching givens and surrounding all polyominoes completed by the givens alone ('1's, pairs of orthogonally adjacent '2's, and so on). From there, the solver searches for three things, possibly in combination: * Potential overloads. Each polyomino in the solution, if it were completely numbered, would contain matching numbers whose quantity is that number. If there is a place in the grid where adding a particular number would result in an orthogonally contiguous region of too many copies of that number, then borders to that cell from those numbers may be drawn in. Often the givens alone provide these, most commonly a pair of ''diagonally'' adjacent '2's: placing a '2' in either of the cells that share a side with both givens would result in an overload, so four cell borders may be drawn in (in the shape of a plus sign) separating the '2's. * Limited domains. Every number in the grid - whether given or deduced - must ultimately be bordered into a region with that number of cells in it. Often, a number will require other cells to be in its region due to not having any alternative location to expand into. The most obvious case is a number (other than '1') bordered on three sides; the cell sharing the fourth side must belong to the same region, and consequently can bear the same number. The same principle applies to numbers bordered on only two sides but cannot possibly expand into enough cells in only one direction, and so on. * Defined cells. In more challenging circumstances, sometimes working with the empty cells is easier than working with numbers. The most obvious case is when a single cell without a number becomes completely surrounded; without any help from other numbers, that cell must be aVariants
Fillomino adapts to different geometries; hexagonal grids can be used, with the only change in the rules being replacing all instances of ''polyomino'' with '' polyhex''. Another variant was published by Nikoli under the name ''NIKOJI''; letters are used as givens instead of numbers, where the letters and polyominoes have a one-to-one correspondence and only matching letters have matching polyominoes (in size, shape, orientation, and letter position). A computer-based variant of the game in which the player enters numbers (the computer draws the boundaries for the player) is known "Filling" and is part of the "SGT-Puzzles" package. AnReferences
{{Reflist Logic puzzles