Fillomino

The trick with Fillomino

Every number you write is really a claim about size. A 4 says it belongs to a block of exactly four cells, no more, no fewer, with the cells joined edge to edge; corners do not count. So the puzzle is less about arithmetic and more about drawing borders. You are carving the grid into connected regions, and the digit inside a region just labels how big that region is. Once that clicks, the given numbers stop looking like scattered clues and start looking like seeds you have to grow to a fixed size.

The rule that does most of the work is the one about touching: two separate blocks of the same number can never share an edge. That single limit builds walls for you. If a finished pair of 2s sits beside a cell that also wants to be a 2, it cannot be. A third 2 there would either swell the block to three cells or start a second 2-block kissing the first, and both are illegal. So that cell holds something else, and you know it for certain.

Ways in

• Start with the 1s. A lone 1 is a complete region by itself, so all four of its neighbours must be a different number. Free walls all around it.
• When a region already has as many cells as its number, cap it off. Every empty cell touching it must belong to a different-sized block, which often forces the digit next door.
• Count escape routes. A half-grown block that can only expand in one direction has to expand that way.
• For a big given near an edge, sketch every cell it could possibly reach. If a 6 can only reach six cells, all six are in, and the whole region solves at a stroke.

A worked deduction

Take the most common small position: two 2s in one row with a single empty cell between them. Test whether that middle cell can be a 2. If it pairs with the left given, the finished two-block would sit edge to edge with the right 2, and equal regions may never touch. Pair it with the right instead and the same crash happens on the other side. Join all three and you have a block of three 2s, one cell too many. Every road fails, so the middle cell is not a 2, and you learned it without touching the rest of the grid. Capping pays the same way. Suppose a pair of 4s has grown to its full four cells: draw its fence at once, because each empty neighbour now refuses the number 4, and if one of those neighbours has only one direction left to grow, the fence just chose its future for it. Most Fillomino solving feels exactly like this, small trials you can run in your head, each ending in a certainty. The habit to build is writing each one down the moment you prove it, because every settled cell shrinks a neighbour's options.

One more thing the rules quietly allow: the finished grid can contain a region whose number was never given anywhere. Solvers call these hidden regions. If the numbers around a sealed three-cell pocket rule out splitting it into a 1 and a 2, the pocket must become a block of 3s even if no 3 appears among the clues. Harder puzzles lean on this on purpose, so never assume the givens are the whole cast.

Where it comes from

Fillomino is one of the pencil puzzles made famous by the Japanese publisher Nikoli, which first ran it in 1994, and it travels internationally under the name Allied Occupation at the World Puzzle Championship. The name stitches together fill and polyomino, the term for shapes built from joined squares, coined by the mathematician Solomon Golomb in 1953 and spread by Martin Gardner's columns in Scientific American. Nikoli's magazine is the same one that turned Sudoku from a filler item into a worldwide habit, and Fillomino is widely rated among its deepest designs. The pedigree fits, because unlike Sudoku there is no ready-made grid of regions handed to you; discovering the regions is the puzzle. Work carefully, trust the walls, and never write a number you cannot yet prove.