Oddsley

One quiet afternoon a young mole climbed the long way up to the oak, puzzled and a little upset. The mole had been practicing at home on an ordinary board, the kind found outside the League, where guessing is sometimes truly necessary. And the mole had become stuck.

"I have two hidden squares left beside this number," the mole explained, holding up a drawing of the board. "Neither one can be proven. My friend says the left one feels safer. But what does it even mean for a square to feel safer? How do you measure a feeling?"

Oddsley hooted softly, and it sounded almost like a laugh. "You don't trust the feeling," she said. "You count the ways."

She glided down from the oak, landed beside the mole's drawing, and lowered her scale onto the grass. "Let's list every single way the mines could be hidden that agrees with all of the numbers on this board. Every arrangement that fits. Not just the ones that feel likely, or the ones that come to mind first. All the ones that are actually possible."

Together, slowly and carefully, they worked through them. Around that 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, it sat on the right square. In just one, it sat somewhere else in the patch entirely. Five arrangements in all. They wrote each arrangement out, one to a line, until there were no more to find.

Oddsley set each arrangement onto her scale like a small brass weight. "Now look closely. The left square has a mine in three of these arrangements. The right square has a mine in only one. If each arrangement is equally likely to be the real one, then the chance of a mine on the left is three out of every five, and on the right only one out of five. The right square is much safer."

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

"The feeling was just a feeling," said Oddsley kindly. "Our minds tend to picture one or two tidy possibilities and then stop, and they can easily picture the wrong ones. Counting doesn't have feelings. It only has arrangements. It doesn't get tired, and it doesn't skip the ones that seem unlikely."

Then she adjusted her spectacles, because she is honest about the fine print. "Now, I did say if each arrangement is equally likely. On a real board, that isn't always quite true. An arrangement that uses more mines and one that uses fewer can't be treated the same, because the board has a fixed total number of mines, and that total changes how likely each arrangement is. Sumsy the hamster knows all about that. She counts the whole jar. But for a small patch like this one, the count we just did gets us most of the way, and far closer than any stomach feeling ever could."

"That's what the Probability Lens does, isn't it?" said the mole slowly. "The one in the League."

"Exactly that," said Oddsley, pleased. "If you turn it on, it shows each hidden square's exact chance of hiding a mine, counting every arrangement the numbers allow, and using the total mine count too, so nothing is left to a hunch. It isn't guessing. It's doing precisely what we just did, only faster and all at once."

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