Oddsley

"Every square has odds," she says in her soft, hooting voice. "Most players feel them. I count them."

One afternoon a young mole came to the oak, confused and a little upset. The mole had been practising on an ordinary board, the kind outside the League where a guess is sometimes needed, and had gotten stuck.

"I have two squares left near this number," the mole said. "Neither one is proven. My friend says the left one feels safer. But how do you know how safe something feels?"

Oddsley hooted softly. "You don't trust the feeling. You count the ways."

She landed beside the board and lowered her scale. "Let's list every way the mines could be hidden that agrees with all the numbers. Every single arrangement that fits. Not the ones that feel likely. All the ones that are possible."

Together, slowly, they worked through them. Around this small patch of hidden squares, the numbers allowed only a handful of arrangements. In some of them, the mine sat on the left square. In some, on the right. In one or two, somewhere else nearby.

Oddsley set each arrangement on the scale like a little weight. "Now look. The left square has a mine in three of these arrangements. The right square has a mine in only one. If every arrangement is equally likely to be the real one, then the right square is safer. Much safer."

The mole's whiskers twitched. "So the feeling was backwards?"

"The feeling was just a feeling," said Oddsley gently. "Counting doesn't have feelings. It only has arrangements."

"That's what the Probability Lens does," she added. "In the League, if you turn it on, it shows you each hidden square's exact chance of being a mine. It isn't guessing. It's doing exactly what we just did: listing every arrangement the numbers allow, and counting."

"But on a League board," said the mole slowly, "I thought you never had to guess."

"That's true," said Oddsley, pleased. "League boards are no-guess, so there's always a proven safe square somewhere. On those boards, the Lens shows you that some squares are exactly zero percent: a mine in none of the arrangements. That's another way of seeing a proof. A zero means safe for certain. The Lens just lets you see the whole picture at once."