Tallis, Bijou and Cammy

Tallis went first, because Tallis had never in her life met a thing she couldn't count twice.

"Watch me," she said. "Each of the seven of us shook hands with the other six. So that's seven people, and six handshakes apiece — seven times six, which is forty-two." She held up a flat hand before anyone could object. "But that number counts every handshake twice over — once from my side of it, and once from yours, because there are two hands in every shake and I've counted both of them. So I cut it in half. Twenty-one handshakes." She sat back, thoroughly satisfied. "Count all the hand-ends, then fix the double. Twenty-one."

Bijou shook her head — not the you're-wrong kind of head-shake, but the that-is-simply-not-how-my-brain-works kind. "I don't count hand-ends at all. I pair things up instead." She pulled a sheet toward her and drew a grid, seven columns across and seven rows down, one column and one row for each friend. "Every handshake is a place where two friends cross — one specific little cell in this grid, where their column meets their row. But the diagonal running corner to corner is you shaking your own hand, which nobody does, so those cells don't count. And every real handshake shows up in the grid twice — once above the diagonal and once below it, mirror images." She carefully shaded only the cells sitting above the diagonal. "So I just count these. The upper triangle. Each shaded cell paired off with exactly one handshake, one to one, nothing left over." She counted them, cell by cell. "Twenty-one."

Cammy, meanwhile, had been grinning through the whole thing.

"You two are doing so much arithmetic," she teased, delighted. "I split it up instead." She tapped the tabletop. "Take the newest friend to walk in the door — that's me; I got here last, remember. Now, every single handshake either has me in it, or it doesn't. Those are the only two kinds. The handshakes without me are just the other six friends shaking hands among themselves. The handshakes with me are me shaking each of those six, one at a time." She wrote it out in two neat piles. "So the grand total is 'however many handshakes six friends make' plus 'my own six.' And here's the lovely part — peel me off the top, and what's left underneath is a smaller copy of the very same question."

She chased it all the way down the staircase: six friends breaking into five-plus-a-newest, and five into four-plus-one, and on and on, adding up each peeled-off layer as she went. "Six, then five, then four, three, two, one," she counted, stacking them. "Twenty-one."

For a second nobody at the table said a single word.

"...We all got twenty-one," Tallis said, slowly, as though testing whether it was safe to believe.

"We all got twenty-one," Bijou agreed, staring at her shaded triangle as if it had turned and winked at her.

Cammy laughed out loud, the sound bright in the quiet. "Well of course we did! It's the same seven people!" She swept a hand over the whole table. "We counted the same exact pile of handshakes three different ways — I split it into layers, Bijou paired it off cell by cell, Tallis doubled it and cut it back down. But it was one single pile the whole time. And however you choose to count one pile, the count is just the count." She grabbed the pen and drew a big, emphatic equals sign linking all three of their answers together. "That, right there, is the whole trick. When two honest ways of counting the same thing come out to different numbers, then one of them has a mistake hiding in it somewhere. But when they come out to the same number — that is not a happy coincidence. That is a proof."

Tallis stared at the equals sign for a long moment. She had spent the entire conversation braced, quietly, to be told her way was the wrong one. Instead the other two roads had come walking up on either side of hers and simply stood there, nodding, agreeing.