Play Fillomino Online

The number is the size of its own block

Every cell in a finished Fillomino grid holds a number, and that number says how many cells are in the block it belongs to. A block of four cells is filled with four 4s. A lone cell is a 1. There is no row or column rule at all — the same number can repeat along a row as often as it likes, as long as each run of it forms a block of the right size.

You start with a few of those numbers scattered on an otherwise blank grid, and with no block boundaries drawn. Working out where the boundaries go is the puzzle; writing the numbers in is just how you record it.

The rule that does the work

One extra rule turns this from bookkeeping into a puzzle: two separate blocks of the same size may not share an edge. If they did they would be joined, and the combined block would be the wrong size for the number in it.

Blocks are smaller than they look

The numbers on screen suggest sprawling shapes, but they do not appear in anything like equal proportion. Across 120 generated boards, more than a quarter of all blocks were a single cell, and two thirds were three cells or fewer.

That skew is what makes a board tractable. A 1 is finished the instant you see it, and it immediately tells you its four neighbours are not 1s. A 7×7 board carries about four single cells and a 10×10 about ten, so there is always somewhere solid to start.

Where to start

Take the 1s. Each is a complete block on sight, and each rules out a 1 in four more cells.

Then look for two identical givens that already touch. They have to be in the same block — they cannot be two separate ones, because separate blocks of a size may not touch. Two adjacent 3s are two thirds of a block of 3 with one cell left to place, and that last cell is often forced. 93% of boards open with at least one such pair already visible, so this is nearly always available.

Then grow the blocks that have only one way out. A given of 3 in a corner whose only free neighbour runs along one edge has no choice. These forced expansions are what carry you off the edges and into the middle.

Count the givens, not just read them

Here is a lever that is easy to miss. On these boards almost every block contains exactly one given — only 2% of blocks have no given inside them at all. A 7×7 Hard board has about 19 givens for about 16 blocks.

So the given count is very nearly the block count, and the numbers themselves must add up to the size of the grid. If your partial solution already needs more blocks than there are givens to anchor them, something earlier is wrong. And two givens reading 4 are almost always two different blocks of four, not two corners of one — they are the same block only if you can already connect them through cells you have assigned.

A worked board

Where solvers get stuck

Completing a block without checking its border. The commonest wrong move is growing a block to the right size and only then noticing it now shares an edge with another block of that size. Check the border before you commit, not after.

Leaving the gaps until last. Blank cells with no given near them feel like they should be filled in at the end. They are often the most constrained cells on the board, because their size is settled by the walls that have closed around them rather than by a number you have to interpret. When a gap is sealed off, count it: a three-cell hole that cannot connect to anything is a block of 3.

Assuming every number you write must already appear. You may write any number you like, given or not. It rarely comes up on these boards — every board measured could be completed using only values that appear somewhere in the givens — but the rule is there, and on a sealed-off gap it is exactly what you need.

Two things you can check about these puzzles

Every puzzle has exactly one solution. This was not previously true. The old generator revealed a fixed share of each block — 60% on Easy, 35% on Medium, 18% on Hard — and never checked what that left behind. Measured against an exhaustive counter, 1 board in 25 had a single solution; the rest had two, three or more, while the page promised “a unique solution solvable with logic alone”. Boards are now built the other way round: reveal one number per block, then keep revealing cells where a rival solution disagrees with the real one until no rival survives, then take back any given that turns out to have been unnecessary. 48 boards across both sizes and all three difficulties were re-checked against a separate counter written from the rules alone. All 48 were unique.

14×14 is gone, and it had to go. The grid-building step never once succeeded at that size in 300 attempts, so every 14×14 board fell through to a fallback tiler that its own comment called “guaranteed valid”. It was not: it dropped in a 1 wherever nothing else fitted, without checking whether that 1 touched another 1, and its output failed the page’s own rule checker at every size. Every 14×14 puzzle ever served here carried an answer key that broke Fillomino’s rules. The size and the fallback are both gone, and nothing now reaches the screen without passing the checker.

Proving a board unique is also why there is a progress bar. A 10×10 takes a few seconds to certify, and doing that work invisibly would just look like a frozen page.

Fillomino and its neighbours

Fillomino is Nikoli’s, and also appears under the name Polyominous. It belongs with the other divide-the-grid puzzles, but it withholds something they hand over. Suguru gives you the regions and asks for the numbers. Nurikabe anchors every island to a given and asks only where the water goes. Fillomino gives you the numbers but not the regions, and asks you to derive one from the other.

That is why counting sealed-off space matters so much here and barely appears in the others: in Fillomino a block can be defined entirely by what has closed in around it.

More puzzles like this

If you like carving a grid into regions, try these: