Fillomino, explained.

Fillomino is a pencil-and-paper logic puzzle from Japan, best known through the puzzle publisher Nikoli. You start with a grid that has a few numbers in it. Your job is to divide the whole grid into regions and write each region's size in every one of its cells.

That sounds loose, but one extra rule makes it tight: two regions of the same size are never allowed to share an edge. Every well-made Fillomino has exactly one solution, and you can reach it by logic alone.

1. Divide the grid into regions. A region is a group of cells joined along their edges. Every cell belongs to exactly one region.
2. Each number is its region's size. A 3 sits in a region of three cells. When the puzzle is solved, every cell shows the size of its region.
3. Equal sizes never share an edge. Two regions of the same size may touch at a corner, but never along a side. Regions of different sizes can touch freely.
4. Givens are only hints. A region can contain one given number, several copies of the same number, or none at all. Regions without a given still follow every rule.

A useful consequence: whenever two cells with the same number touch along an edge, they are in the same region.

Most Fillomino solving comes down to a handful of moves, used again and again. The boards are drawn the way Slime Squares draws them: each slime is one region, and its colour shows its size. Yellow bars are borders you have just worked out, and dashed outlines are newly filled cells.

1. 1Ones are finished from the start

   A 1 is a region of one cell, so it is already complete. Draw a border on all four sides: none of its neighbours can join it, and none of them can be a 1 either.

2. 2A 2 with one way out

   This 2 sits in the corner of the board. It needs exactly one more cell. Its right-hand neighbour is a finished 1, so the only cell left is the one below it.

3. 3Equal numbers that touch are one region

   Two 3s side by side share an edge, and equal regions may not touch along an edge, so these two cells must be the same region. It needs exactly one more cell, and no other 3 may join it.

4. 4Keep equal numbers apart

   The empty cell between these 2s cannot join either one. If it joined the left 2, that finished region would touch the right 2 along an edge. The same goes the other way, and joining both would make a region of three. So the middle cell belongs to a different region.

5. 5Too big if joined

   The 4 on the left already has three cells. If the gap were a 4, it would connect them to the other 4 and make a region of five, which is too big. So the gap is not a 4, and the left region must find its fourth cell somewhere else.

6. 6Fill sealed pockets

   In this corner of the board, two empty cells are surrounded by finished regions, so they must form their own region. Two separate 1s would touch each other, which is not allowed, so the pocket is a single region of size 2.

When none of these applies, count reach: for each unfinished region, ask which empty cells it could still grow into, and for each empty cell, ask which regions could still reach it. A cell that only one region can reach belongs to that region.

Five beginner puzzles, roughly easiest first. Each one was checked by a solver to have exactly one solution. Print the page or copy them onto paper, and open a solution only once you are done.

1. Puzzle 1

   Show solution

2. Puzzle 2

   Show solution

3. Puzzle 3

   Show solution

4. Puzzle 4

   Show solution

5. Puzzle 5

   Show solution

Slime Squares is a free puzzle game built on Fillomino's core rule: each number is the size of its region, and equal sizes never share an edge. It changes one thing. Every region, called a colony, has exactly one number: the slime it grows from. You never hunt for regions without a given, so the puzzle is about shapes and edges, and you solve it by dragging slimes to grow them.

It adds rocks and shaped boards, 500 levels from tiny tutorial gardens up to 8 × 8 boards, a new Daily Square every day, and hints that explain the next deduction. The Slime Squares rules page walks through a full example.