The Pizza That Could Not Win

There were eleven kids in the Maker Club and three kinds of pizza on the whiteboard. Mushroom. Pepperoni. Pineapple. One pizza only, because the budget was one pizza only, and the meeting was supposed to last ten minutes.

"Easy," said Maya. "Everybody raise a hand for their favorite."

Soren counted. "Pepperoni, four. Mushroom, four. Pineapple, three."

"Tie," said Maya. "Okay. New plan. Rank them. First choice, second choice, third choice. Write it down."

Soren collected the slips and spread them on the table like a hand of cards. He liked seeing all of it at once.

"Pineapple has the fewest first-place votes," he said. "Knock it out. Then see who the pineapple people liked second."

They did. All three pineapple voters had put mushroom second.

"Mushroom wins," said Soren. "Seven to four."

"Wait." Maya was already pulling the slips back toward herself. "Do it the other way. Add up everybody's points. Three points for first, two for second, one for third."

Soren ran the numbers twice because he did not trust them the first time.

"Pepperoni," he said slowly. "Pepperoni wins on points."

They looked at each other.

"Same slips," said Maya.

"Same slips," said Soren. "Same eleven people. Nobody changed their mind."

Nadia, who ran the club and was mostly busy untangling a robot's wiring harness in the corner, called over without looking up. "Just pick one. It's pizza."

"It's not pizza," said Maya. "It's wrong. The same votes can't have two winners."

"Then there's a right way to count and we picked the wrong way," said Soren. "Find the right way."

So they tried to find it.

They tried head-to-head. Pepperoni against mushroom, just those two, ignore pineapple for a second.

"Mushroom beats pepperoni, six to five," Soren said.

"And mushroom against pineapple?"

"Mushroom, seven to four."

"So mushroom beats both." Maya grinned. "Mushroom is the real winner. It wins every fight."

"Then why did pepperoni win on points," said Soren.

Maya stopped grinning.

"Because points let pineapple's third-place votes leak in somewhere," she said. "The losing pizza changed who won between the other two."

Soren went quiet, then pulled a fresh slip and drew three rules on it, the way he drew steps for a circuit.

"Okay. What do we even want from a fair count? Say it out loud and I'll write it."

Maya paced. "One. If everybody likes mushroom better than pepperoni, the system can't pick pepperoni. Obvious."

"Got it."

"Two. Pineapple lost. Pineapple losing shouldn't change whether mushroom or pepperoni wins. It's out. It shouldn't matter."

Soren wrote it. "That's the one the points method broke."

"Three." Maya slowed down. "No single person gets to decide for everyone. No dictator. We actually count everybody."

Three rules on the slip. They both stared at them. They looked completely reasonable. They looked like nothing anyone could argue with.

"Now build a counting method that does all three," said Maya. "Go."

They went.

They tried knocking out the lowest first. That broke rule two, because if they knocked out mushroom instead of pineapple the whole answer flipped, and which one got knocked out depended on pineapple sitting there doing nothing.

They tried points again. Rule two, broken, same way.

They tried head-to-head all the way around. Mushroom beats pepperoni. Pepperoni beats pineapple. Pineapple beats mushroom.

Soren put his pencil down. "That's a loop."

"A loop," Maya repeated.

"Mushroom beats pepperoni beats pineapple beats mushroom." He traced the circle with his finger. "There's no top. It's a circle with no top."

Maya leaned over the slips, breathing through her mouth "Try another method," she said. "Any method. We just haven't found it."

They tried for twenty minutes. The robot got untangled. Nadia ate a granola bar and offered them half and they didn't hear her.

Every method they built picked a winner. And every single one, when they pushed on it, broke one of the three rules on Soren's slip. Knockout broke rule two. Points broke rule two. Letting one kid choose satisfied rules one and two perfectly and broke rule three, the dictator one, instantly. They could always get two of the rules. They could never get all three.

"It's not us," Maya said finally. She said it the way you say something you've checked six times. "We're not bad at this. It can't be done."

Soren looked up. "You can't know that. We tried like nine methods."

"There's infinity methods, Soren. We can't try them all." She pointed at his slip. "But every method has to do something with those three rules. And every time we point a method at the rules, one rule falls over. Not because the method's dumb. Because the rules can't all stand up together. Once there's three pizzas."

Soren stared at the three rules. Then at the loop. Mushroom, pepperoni, pineapple, back to mushroom.

"Two pizzas would be fine," he said slowly. "Two pizzas, majority wins, done. No loop."

"Right."

"It's the third thing." He tapped pineapple. "The third choice is what makes it impossible. The crowd doesn't have a favorite. It has a circle."

The room felt suddenly bigger to Maya, the way a room does when you find out the wall you were leaning on opens into another room.

"Every election," she said. "Every real one. President. Class president. Everything with three or more choices."

"Has a hidden circle in it somewhere," Soren said. "And whoever picks how you count is picking which rule gets broken."

"And there's no count that breaks none." Maya's voice had gone very low. "Somebody figured this out already. They had to. This is too clean."

"Somebody proved it," said Soren. "You can prove a thing can't exist. We just did it for pizza."

Nadia wandered over, brushing crumbs off her hands. "Did you guys pick yet? It's been forty minutes."

Maya looked at her. "Whatever we order, the math says some of us got cheated and there was no way not to."

Nadia laughed. "It's pizza."

"It's everything," said Soren.

Maya picked up the three slips of paper, the knockout result and the points result and the loop, and laid them side by side facing Nadia.

"Pick which rule you want to break," she said. "That's the only choice there is."

Nadia stopped laughing and looked down at the three pieces of paper for a long time.