Sheets of paper covered in mathematical calculations, most of them crossed out, spread over the table and cascaded on to the floor. Mathophila picked up one of them and looked at it.
"You've had an idea, then," she said.
"I've made a discovery," Innumeratus said. "If you arrange the numbers from 1 to 4 in the order 3 1 2 4, then in each pair of consecutive numbers, one is an exact multiple of the other."
"You mean 3 is a multiple of 1, and 2 is a multiple of 1, and 4 is a multiple of 2?" asked Mathophila.
"Yes," said Innumeratus, "But I don't think you can do it with the numbers from 1 to 5. I've tried, but nothing works."
"No, that can't work," said Mathophila. "The only number that can follow or precede 3 is 1, and the same goes for 5. So you get 3 1 5 or 5 1 3, but there's no way to include 2 and 4."
"Well, I can do it with the first six numbers—5 1 4 2 6 3," said Innumeratus. "But I suppose that when there are six numbers, 3 can be followed or preceded by 6 as well as 1."
"I wonder if there are longer sequences of this kind," said Mathophila. "Hey! I can do it with the numbers from 1 to 12!"
Innumeratus peered at her list. "I think you've missed out 11."
"So I have. Still, you have to admit I came close."
What was Mathophilia's sequence?
Prospect invites you to solve the puzzle and send us the solution. Correct answers will be entered into a draw. The winner will receive Impossible: Surprising Solutions to Counterintuitive Conundrums by Julian Havil (Princeton University Press, £16.95).
Send answers to info@prospect-magazine.co.uk by 12th June. The winner will be announced in our July issue.
