Sumsy

One afternoon Oddsley the owl and a young mole were puzzling over the end of a board. They had been carefully counting the ways the mines could be arranged around the last few numbers, just as Oddsley had taught. But something didn't add up.

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

Sumsy rolled over, jar held tight. "Not anything," she said. "Have you counted the jar?"

The mole blinked. "The jar?"

"This board has ten mines in total," said Sumsy, holding up her jar. "That's the whole jar. So let's count. How many mines have you already found?"

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

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

Oddsley looked back at the arrangements they had listed. "Some arrangements put three mines around those numbers. Some put four."

Sumsy's whiskers twitched with excitement. "Then here's the magic. The jar has ten. Six are found. That leaves four more, somewhere. If an arrangement puts four mines around the numbers, then zero mines are left for the corner patch. If it puts three, then exactly one mine has to be hiding in the corner."

The mole's eyes went wide. "So the corner isn't a mystery at all. It depends on the numbers."

"It depends on the jar," said Sumsy happily. "Every arrangement has to use up the jar exactly. Not one pebble more, not one pebble less. That rules out lots of arrangements that looked possible before. And when some arrangements are ruled out, the odds of every square shift, even squares no number touches."

Oddsley tilted her head approvingly. "The Probability Lens uses the jar too," she said. "That's why its odds are exact. It counts every arrangement that fits the numbers and the total."

"So the total is like one giant extra number," said the mole slowly, "that touches the whole board at once."