Tallis, Bijou and Cammy
THREE METHODS, ONE IDENTITY — a combinatorial proof is trusting that counting the same collection different ways must agree: Tallis counts every handshake twice (once per hand) and halves it to n(n-1)/2, Bijou matches each handshake one-to-one with a cell above a grid's diagonal, and Cammy splits the handshakes by whether the newest friend is in them — and all three totals are forced to be equal.
A story read by Tallis, Bijou and Cammy
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There were seven of them at the table, and they'd just finished a round of hellos — everyone shaking hands with everyone else, once each. Now Tallis wanted to know: how many handshakes was that, exactly?
"Easy," said Bijou.
"Easy," said Cammy, at the same time.
"Easy," said Tallis — and then all three of them said a different plan for how to count it, and stopped, and eyed each other. Each was suddenly, quietly sure that their way was the real way, and that the other two must be about to make a mistake.
Tallis went first, because Tallis never met a thing she couldn't count twice.
"Watch," she said. "Each of the seven of us shook hands with the other six. So that's seven people times six handshakes each — forty-two." She held up a hand before anyone objected. "But that counts every handshake twice — once from my side and once from yours. So I halve it. Twenty-one handshakes." She sat back, satisfied. "I counted the hand-ends, then fixed the double."
Bijou shook her head — not to say wrong, but to say not how I'd do it. "I don't count hand-ends. I pair things up." She drew a grid, seven columns and seven rows, one for each friend. "Every handshake is a spot where two friends cross — a cell in this grid. But the diagonal is you-shaking-your-own-hand, which isn't a thing, and each real handshake shows up twice, above and below the diagonal." She shaded the cells above the diagonal only. "So I just count these — the upper triangle. One-to-one with the handshakes." She counted. "Twenty-one."
Cammy had been grinning the whole time.
"You're both doing so much arithmetic," she teased. "I split." She tapped the table. "Take the newest friend to arrive — that's me, I got here last. Every handshake either has me in it, or it doesn't. The ones without me are just the other six shaking among themselves. The ones with me are me shaking each of the other six." She wrote it out. "So the total is 'six friends' worth of handshakes' plus 'my six.' Peel me off, and you've got a smaller version of the exact same question underneath."
She chased it down — six friends split into five-plus-a-newest, and five into four-plus-one — adding up the peeled-off layers: six, then five, then four, three, two, one. "Twenty-one," she finished.
For a second nobody said anything.
"...We all got twenty-one," Tallis said slowly.
"We all got twenty-one," Bijou agreed, looking at her shaded triangle as if it had winked at her.
Cammy laughed out loud. "Of course we did. It's the same seven people! We counted the same pile of handshakes — I split it, Bijou paired it, Tallis doubled-and-halved it. It's one collection. However you count it, the count is the count." She wrote a big equals sign between their three answers. "That's the whole trick. When two honest ways of counting the same thing land on different numbers, one of them has a mistake. When they land on the same number — that's not a coincidence. That's a proof."
Tallis stared at the equals sign. She'd spent the whole time bracing to be told she was wrong. Instead, the other two paths had walked up beside hers and stood there, agreeing.
A younger kid had been watching from the doorway, the way kids do when a table of older ones is arguing about something that turns out to be friendly.
"So which way is the right way?" the kid asked.
The three friends looked at each other.
"All of them," Bijou said. "That's the point. There isn't one right road. There's one right place, and a bunch of roads to it — and when the roads all arrive at the same place, you know you're really there."
"Pick the road that feels like yours," Tallis added. "I like counting twice and fixing it. Bijou likes matching things up. Cammy likes peeling off a layer."
"And when we compare," Cammy said, "and the numbers agree — that agreement is the proof. Count the same thing two ways, and the two totals have to match. If they match, you're done."
The kid looked at the grid, and the doubled forty-two, and the peeled-off layers, and the single stubborn twenty-one sitting under all three.
"So I could check my own answer," the kid said, "by counting it a different way. And if they agree, I don't have to wonder anymore."
"Now you're counting," said Tallis.
The CountCraft ensemble
Tallis, Bijou and Cammy is part of CountCraft's distributed-narrative cast. Each character embodies a different curricular primitive; together they teach the full subject.
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Tallis
The Two-Ways Counter — counts the same collection along rows, then along columns, and insists the two totals be equal (double counting)
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Bijou
The Pair-Matcher — draws a one-to-one pairing between two sets to prove they are exactly the same size (bijection)
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Stacia
The Staircase-Mirror — mirrors a one-two-three staircase against itself to make a rectangle (triangular numbers)
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Nona
The Gnomon-Grower — adds the next odd L-shaped layer to grow a square one ring at a time (perfect squares)
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Arel
The Square-Cutter — cuts a square into four rectangles to read off a-squared plus two-a-b plus b-squared (area model)
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Domina
The Domino-Tiler — tiles a strip with squares and dominoes and counts the number of ways (Fibonacci tilings)
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Chessa
The Handshake-Counter — pairs everyone with everyone exactly once to count the handshakes (choose two)
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Cammy
The Committee-Splitter — sorts every committee by whether the last person is in or out (Pascal's rule)
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Sett
The Subset-Flipper — flips an in-or-out coin for each item to list every subset exactly once (two-to-the-n)
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Parry
The Parity-Prover — two-colours a board to show when a tiling simply cannot exist (parity and impossibility)
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Quill
The Proof Director (mentor) — a warm, attentive adult who coaches build-the-two-counts-then-read-the-identity, and never proves for the kid
