The pun is so obvious, I'm surprised no one else has used it as a name for this puzzle. But then again, maybe that's exactly why they haven't.
A polyomino is a tile made of a group of squares. The video game Tetris gets its name from the pieces being tetrominoes - that is, four-unit polyominoes. A two-unit polyomino is called a domino, a term most should be familiar with; a one-unit polyomino is called a monomino. For those who want to hum along without learning the words, a polyomino of n units can be called an n-omino, so a tetromino can be called a 4-omino, a polyomino with fourteen cells is a 14-omino, and so on.
On the left is an unsolved Polyominous puzzle. On the right is what its unique solution would look like if solved with purple ink.
The object is to take the given grid and divide it into polyominoes such that each given number n in the grid is part of an n-omino and that no two polyominoes of matching size (quantity of cells) are orthogonally adjacent.
Yes, that's the entirety of the rules in one sentence. It's a simple puzzle. But for the checklist lovers out there:
1) Except for the outside border, the puzzle grid shows only the corners of the cells of the grid. To solve the puzzle, draw in the needed borders between cells so as to divide the grid up into sections following the remaining rules.
2) The number of squares in each section created by the borders must match the number in any cells of that section. (So if there's a 2 in the grid, the section with that 2 must have exactly one other cell with it, making two cells total.)
3) Two sections with the same number of cells cannot be orthogonally adjacent - that is, they may not touch except at corners. It is strongly recommended that as you learn what cells belong to what sections, you write in numbers in the blank cells - just like those already in the grid - so that you don't violate this rule. (So if you have a three-cell section, you can't have another three-cell section sharing a cell side with it. If you write a '3' into each cell of the section that needs it, you'll know not to write another '3' next to it as part of another section.)
Note that this isn't Islands in the Stream. The numbers in the grid are not necessarily one-to-a-polyomino. You can have multiple given numbers in the same section, like the triomino (3-omino) in the upper right of the sample, or even no digits at all, like the monomino in the upper left of the sample. You'll need to pay careful attention as you're divvying up the grid. Don't settle for trial and error - stick to logical deduction.
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This was an interesting one to put together. I had some fun with patterns in the given numbers. It starts out easy enough, but I recommend caution as it goes along. Comments are welcome as always, and emailing me your solution might get you something. Note that a simple grid of numbers, if you fill in all the blanks, is sufficient to express the answer; the borders are trivially deducible from that. - ZM