The Bridge That Wouldn't Close

The festival map had a red stamp box at the bottom that said: CROSS ALL SEVEN BRIDGES ONCE. NO BRIDGE TWICE. GET YOUR CARD PUNCHED AT EACH. WINNERS GET THE BIG PRIZE.

"We have until the fireworks," Maya said. "That's hours. This is easy."

"I don't think it's easy," said Soren. He was looking at the map upside down, then right side up. "I think it's weird."

They started at the north bank. The river split around two islands in the middle of town, and seven old iron footbridges stitched everything together. Bridge one, punch. Bridge two, punch. They crossed to the big island, crossed to the little island, doubled back.

An hour later they were standing on the north bank again, sweaty, and Maya's card had six punches on it.

"One left," she said. "The east bridge."

"But we're on the wrong side," Soren said. "The east bridge starts from the island. To get to the island we have to cross a bridge we already used."

"So we cross it again."

"Then it's twice."

Maya frowned at her card. "Okay. New plan. We start over."

They started over. Different first bridge, different order. This time they left the east bridge for the very end again and ended up trapped on the little island with one clean bridge stranded across the water where they couldn't reach it without repeating.

They started over a third time. A fourth.

A man selling roasted corn watched them stomp past his cart for the fifth time. "Nobody wins that one," he said cheerfully. "Been running the festival forty years. They put the prize up because nobody ever claims it."

"Somebody has to be able to," Maya said.

"Suit yourself," the corn man said, and turned back to his grill.

Soren had stopped walking. He had his notebook out and he was not drawing the river. He was drawing dots.

"What are you doing," Maya said. It wasn't really a question. She came and stood over his shoulder.

"The distances don't matter," Soren said slowly. "How long the bridges are. Where exactly they go. None of it matters. The only thing that matters is which piece of land connects to which piece of land."

He drew four dots. North bank. South bank. Big island. Little island.

Then he drew the bridges as lines between the dots. Two lines from the big island to the north bank. Two from the big island to the south bank. One to the little island. And the little island connected to north, south, and big island, one bridge each.

Maya went quiet. She put her finger on the big-island dot and counted the lines coming out of it.

"Five," she said. "Five bridges touch the big island."

"Little island," Soren said, counting. "Three."

"North bank, three. South bank, three." Maya sat down right there on the cobblestones. "Soren. Every single dot has an odd number of bridges."

"Is that bad?"

"Think about a dot in the middle of your walk. Not where you start, not where you finish. You walk in on one bridge, you have to walk out on another. In, out. In, out. They come in pairs." She was talking fast now. "If a place is in the middle of your walk, it needs an even number of bridges. One to arrive, one to leave, every time."

Soren stared at the dots. "So an odd dot can only be a start or an end."

"Because at a start you leave without arriving. At an end you arrive without leaving. Those are the only two spots that get to be odd."

"A walk has one start and one end."

"Two odd dots allowed. That's the whole budget. Two." Maya looked up at him. "How many odd dots do we have?"

Soren counted them again even though he already knew. "Four."

Neither of them said anything for a second. The river kept moving underneath all seven bridges, not caring.

"It's not that we're bad at it," Maya said. "It's not that we haven't found the right order yet."

"There is no right order," Soren said. "There can't be. Not for any person, ever, no matter how long the festival runs." He looked at his four dots and four lines and felt the town rearrange itself in his head, all the iron and stone of it collapsing down into a picture you could hold in one hand. "The corn man was right. But he didn't know why."

Maya took the notebook. She traced the lines with her finger, not the streets, the lines. "You could cover the whole world like this," she said. "Everything that connects to everything. Roads. Phone calls." She looked up. "Who's friends with who. It'd all just be dots and lines."

"And some of them would be possible to walk," Soren said, "and some of them wouldn't, and you could know which just by counting. Without leaving your chair."

"Without even seeing the real thing." Maya laughed, a little shaky. "You don't need the bridges. You need the dots."

They walked back to the stamp table together. The festival woman there had a clipboard and a bored expression and a jar of little pencils.

"Six punches," she said, looking at Maya's card. "You need seven. Prize is seven."

"Nobody can get seven," Maya said.

"Rules are rules, hon."

"No, listen." Maya put Soren's notebook on the table, right on top of the clipboard. "Nobody can get seven. Not because they're not trying. Because it's impossible. There are four places with an odd number of bridges, and a walk like this only allows two. You can check it yourself. It's not a hard puzzle. It's a proved thing."

The woman looked at the dots. She looked at them for a long time, longer than a bored person looks at anything. She picked up one of the little pencils and touched each dot, counting under her breath.

"Huh," she said. "Forty years."

She reached under the table, past the big prize box, and came up with a second box, dusty, unopened, with a faded label that said FOR ANYONE WHO PROVES IT CAN'T BE DONE.

She blew the dust off and set it in front of them.

"Somebody printed this card a long time ago," she said, "and I always figured they were joking."

Maya and Soren looked at the box, and then at the four dots, and then across the water where all seven bridges stood exactly where they had always stood, connecting exactly what they had always connected, waiting for nobody.