Floody

A young hedgehog came trundling down to the pond one afternoon, his quills slightly on end with confusion. "Something weird keeps happening," he said. "Sometimes I click one single square, and a huge chunk of the board just opens up all by itself. Half the board, sometimes. Is the game broken?"

Floody let out a big, honking, happy laugh that sent ripples across the pond without any help from her wings at all. "Broken? Not a bit! That's the very best thing in the whole game. Come here, I'll show you exactly why."

She waddled out onto a fresh board and pointed across it with one wingtip. "Every square you open shows a number, and that number tells you how many mines are touching it, in the eight squares all around. A one means one mine is somewhere next to it. A two means two. And a zero..." She paused dramatically, eyes sparkling. "A zero means that none of the eight squares around it hides a mine. Not a single one."

"So?" said the hedgehog.

"So if no mines touch a square," said Floody, "then every square next to it must be safe. All eight, guaranteed. The game knows that just as well as you do, so instead of making you click them one by one, it opens all of them for you. Automatically. You don't lift a claw."

She tapped a square with her bill. It showed a zero, and at once the eight squares around it flipped open.

"And now here's the splashy part," she went on, bouncing on her webbed feet. "Look at those eight new squares. If any of them is also a zero, then its neighbors must be safe too. So the game opens those as well. And if any of those are zeros, it opens their neighbors, and on and on."

The hedgehog's eyes grew wider and wider as he watched. The opening spread outward from zero to zero, square to square, rolling across the board in exactly the way Floody's wave rolled across the pond. It kept going and going, until finally it came up against a ring of squares that each showed a number, squares that touched at least one mine. And there, all around its edge, it stopped.

"It stops at the numbers," the hedgehog said, amazed.

"Exactly right," said Floody. "The numbers are the shore. A square with a number on it isn't a zero, so it doesn't spread any further. It gets opened, because it sits right beside a zero, but its own neighbors stay closed. So the wave washes open the entire empty region, every connected zero, and then it stops neatly along a border of numbers, the way water stops at the edge of a pond. People who write computer programs call this a flood fill, because that's exactly what it looks like: something pouring outward until it hits a wall. And all of it," she spread her wings as wide as they would go, "happened from one single click."

"That's why Bevvy counts each empty region as only one click," the hedgehog said slowly, remembering what the honeybee had taught him. "Because one splash opens the whole thing."

"Yes!" quacked Floody, so pleased she did a small spin. "An entire empty region, however enormous it is, only ever costs one click, because one zero sets the whole chain going. That's why finding a zero feels so lucky and lovely. One little tap and a huge part of the board opens up to you, and every one of those new numbers along the shore is a fresh clue."