Knowsley and Oddsley

They argued about it for a whole season. Then one evening, on a hard board at the far end of the League Field, they found out they had been saying the same thing all along.

A young porcupine named Thistle had asked them both for help. She was halfway through a Club board, with a stubborn patch of hidden squares near a cluster of ones and twos. "I don't know what to do next," she said. "Can you show me?"

Knowsley answered first, of course. "Let's find a proof." He studied the numbers, tapping his temple, and soon pointed to a one and a two sitting side by side. "Every hidden square of the one is also next to the two. The one says one mine is in those shared squares. The two needs two. So the two's extra square must hold the other mine. Proven."

Thistle flagged it, impressed.

Then Oddsley swooped down beside them, head tilted all the way sideways. "Now let me show you the same patch with my scale." Together, she and Thistle listed every arrangement of mines that fit all the numbers, and the total from Sumsy's jar too. There weren't many. Oddsley weighed each one.

"Look at the square Knowsley just flagged," Oddsley said. "In how many arrangements is it a mine?"

Thistle counted. "All of them. Every single arrangement has a mine there."

"And that square over there," Oddsley pointed with a wingtip. "In how many is it a mine?"

"None," said Thistle slowly. "Not one."

Oddsley turned to Knowsley, eyes twinkling behind her spectacles. "A square that's a mine in every arrangement," she said, "is a mine you've proven. A square that's a mine in none of them is a safe square you've proven. My scale finds your proofs too. They're just the squares at a hundred percent and zero percent."

Knowsley stopped tapping. He looked at the scale for a long time. "So when I find a proof," he said, "I'm finding a square where every possible way agrees."

"Yes," said Oddsley.

"And when you weigh odds, the squares in between are the ones where the arrangements disagree."

"Exactly," said Oddsley. "On a League board, there's always at least one square at zero or a hundred. That's why you never need to guess. Your proofs and my odds are one idea, seen from two sides."

Thistle opened the zero-percent square. Safe. A new number appeared, and with it, new proofs. The patch unravelled.