Sumsy

One afternoon, Oddsley the owl and a young mole were puzzling together over the very end of a tricky board. They had been doing exactly what Oddsley teaches: carefully listing every way the mines could be arranged around the last few numbers. But something about it didn't add up, and they both knew it.

"There's a patch of hidden squares down in the corner," the mole said, pointing, "far away from every number. No number touches them at all. So there's no clue about them whatsoever. They could be anything. There's no way to reason about them."

Sumsy came rolling over, jar held tight to her chest. "Not anything," she said brightly. "Have you counted the jar?"

The mole blinked. "The jar?"

"This board has ten mines in total," said Sumsy, holding her jar up so the pebbles clinked. "It says so right at the top of the board, before you ever start. That's the whole jar. So let's count properly. How many mines have you already found and flagged?"

They counted the flags together. "Six," said the mole.

"And around those last few numbers, how many mines do your arrangements need?"

Oddsley looked back over the arrangements they had so carefully listed. "Some of them put three mines around those numbers," she said. "Some of them put four."

Sumsy's whiskers quivered with excitement. "Then here's the magic part," she said. "The jar holds ten. Six are already found. So exactly four more are still hiding, somewhere on the board. Now, if an arrangement puts four mines around the numbers, then there are zero left over for the corner patch. It has to be completely empty. But if an arrangement puts only three mines around the numbers, then exactly one mine must be hiding down in the corner. It's the only place left for it to go."

The mole's eyes went very wide. "So the corner isn't a mystery at all. What's in it depends on what's around the numbers."

"It depends on the jar," said Sumsy happily. "Every possible arrangement has to use up the jar exactly. Not one pebble more, not one pebble less. That rules out lots of arrangements that looked perfectly possible when you only looked at the numbers. And there's something even subtler. The corner patch has several squares, so an arrangement that leaves one mine for the corner can place that mine in any of those squares. That means it really comes in several versions, one for each corner square, while an arrangement that leaves the corner empty comes in only one. So the arrangements aren't all equally weighted anymore. The jar changes how much each one counts. And when that happens, the odds of every single square shift, even squares no number ever touches."

Oddsley tilted her head in slow approval. "This is why the League's Probability Lens is exact," she said. "It doesn't only count arrangements that fit the numbers. It counts arrangements that fit the numbers and use up the jar exactly, and it weighs each one by how many ways the leftover mines could be spread across the far-off squares. That's the fine print I always warn people about."

"So the total is like one giant extra number," the mole said slowly, working it through, "one that touches the entire board at once, every square, even the lonely corner ones."