Knowsley

Knowsley padded over and sat down beside her in the wet clover. He looked at the board for a long, quiet moment. Tap, tap, tap.

"Do you know what's special about the boards in the League?" he asked gently.

She shook her head.

"Every League board is built to be no-guess," he said. "Before a board is ever handed to anyone, a checking program plays it through by logic alone, the way a very patient player would. It starts from the free opening and asks, at every step, can the numbers prove at least one more square is safe, or prove one more square is a mine? If ever the answer is no, if the checker ever gets stuck with no provable square, that board is thrown away and a new one is made. So every board you ever receive has been proven solvable, one logical step at a time, from the first opening all the way to the last square. Every one. No exceptions."

The rabbit frowned up at him. "But I'm stuck right now. On this exact board."

"Then here's the wonderful thing," said Knowsley, his voice warm. "If this board is no-guess, and you're stuck, then being stuck can't possibly mean you have to guess. It has to mean there's a proof sitting on this board that you haven't spotted yet. The answer is definitely there. The checker found it. It's just hiding from you."

That single idea changed the whole board for her. It wasn't a wall anymore. It was a puzzle with a guaranteed solution, and a guaranteed solution is something you can hunt for with hope instead of dread.

"So where is it?" she asked, sitting up a little.

Tap, tap, tap. "Let's go looking for the trickier kind of proof," said Knowsley. "The easy proofs use just one number at a time. But sometimes no single number can prove anything by itself. Then you have to look at two numbers together, and see what they say when you put them side by side."

He pointed with one paw to a one and a two sitting next to each other. "Look carefully at their hidden neighbors. Every hidden square touching this one is also touching the two. The two's hidden squares include all of the one's, plus one extra square the one doesn't touch. Do you see?"

The rabbit leaned in. "Yes. The two has the same squares as the one, and then one more."

"Now think it through," said Knowsley. "The one says there's exactly one mine among its squares. Those squares are all inside the two's squares as well. So that one mine counts toward the two's total too. But the two needs two mines. One of them is already in the shared squares. So where's the other one?"

"It can't be in the shared squares," the rabbit said slowly, working it out, "because the one only allows one mine there. So it has to be in the extra square, the one only the two can see!" Her ears shot up. "It's a mine!"

"It's a mine," Knowsley agreed, smiling broadly. "Proven, with no guessing at all. Players sometimes call this the subset rule, because one number's squares sit completely inside another's. Two numbers together did something that neither one could ever have done alone."

She flagged the square with a careful paw. And the moment she did, a neighboring number was satisfied, which proved a new square safe. She opened it. It showed another number, which proved another square. And suddenly she wasn't stuck at all. The board unfolded in front of her, proof after proof, until the last hidden square flipped over and the whole board was clear.