Twinnie

Some time ago, two badger cousins arrived at the lab in the middle of an argument that had clearly been going for days. The first cousin, Pim, insisted you should never take a card back down once it had gone up to the finished foundation piles. It felt like undoing your own progress. The second cousin, Mossy, insisted that taking cards back was a clever trick that won far more games. Neither would budge, so they agreed to settle it with a contest.

Pim played ten games his way. Mossy played ten games her way. When they were done, Mossy had won six. Pim had won three.

"There," Mossy said, folding her arms. "My rule wins twice as often. It's settled."

Twinnie had been watching the whole contest from her shelf, whiskers twitching. Now she slid down, landed on the floorboards with a soft wet thump, and asked a question that made both cousins go completely quiet.

"Were your twenty deals the same?"

They weren't, of course. Pim had shuffled and dealt his own ten. Mossy had shuffled and dealt hers. And solitaire deals vary enormously from one shuffle to the next. Some come out generous: the aces near the top, the long descending runs falling neatly into place, every card you need arriving just when you need it. Others come out hopelessly tangled, with every useful card buried beneath a King and no way to dig it free. A tangled deal cannot be won under any rule. A generous one can be won under almost any rule.

"So maybe Mossy's rule really is better," Twinnie said. "Or maybe Mossy just happened to be dealt the generous hands while Pim got the tangled ones. From your contest, there is no way to tell which. The luck of the deal and the quality of the rule are mixed together like two colors of paint in one jar. You can't pour them back apart."

Pim looked relieved. Mossy looked annoyed. And Twinnie looked like an otter with a plan.

She took a single fresh deck, shuffled it thoroughly, and dealt twenty hands. Then she did something slow and very careful. She dealt those exact same twenty hands again, card for card, into a second row directly beneath the first. It took her a long time, and she checked every card twice. When she was finished, there were two identical rows of twenty deals, each deal in the top row matched precisely by its twin below. She pushed the top row toward the solver to be played under Pim's rule, and the bottom row to be played under Mossy's.

Now every deal had a twin. Deal number seven for Pim was exactly deal number seven for Mossy. If a deal was generous, it was generous for both rules. If it was tangled, it was tangled for both. The luck of the shuffle was identical on both sides, so it could no longer tip the scale either way. Whatever difference showed up now could only have come from the one thing that differed: the rule.

This is what scientists call a paired test, and it is one of the most powerful tricks in the lab. By making the two conditions share the same luck, you let the luck cancel itself out.

When the results arrived, the dramatic gap between the cousins had almost vanished. On most deals, both rules won, or both rules lost, because the deal itself decided the outcome. Only a handful of deals came out differently between the two rows.

"Those few," Twinnie said, tapping them one at a time, "are the only places where the rule actually made a difference. Everything else was just the deck talking."

Mossy was silent for a long moment, studying the two rows. "So my rule is a little bit better," she said at last, slowly. "Not twice as good."