Stefano Splodji-Ryannou had bought an airline. "I'm calling it Stefano Planes," he announced to his staff.
"Sorry, can't do," said the advertising executive. "With the big bold letters we use, it's too long to fit on the aircraft."
"Well… my grandma called me Fano, for short. Use that instead. We'll fly people from London to Lanzarote for £1." He paused. "Plus taxes, food, and of course oxygen."
"By 'London' you mean that little airport in the Orkneys?"
"Right. Plus local ground transport. And sea transport from Tierra del Fuego to Lanzarote."
"We have landing rights at seven cities," said the operations manager. "We could use one of them as a hub."
"No, our marketing surveys show that passengers don't like hubs—they don't want to change planes. So I've come up with a scheme. I want each plane to fly a triangular route linking three cities. And I want every pair of cities to be on exactly one route, for maximum efficiency."
The operations manager thought carefully. "Depends on how many planes we've got," he said. "With seven cities, there are seven possible starts multiplied by six possible destinations—42 combinations. Now, each route will handle six combinations. So the minimum number of planes that we need is 42 divided by six—seven."
"By a strange coincidence," said Stefano, "that's exactly how many we own."
"That helps. But with the constraints you've given us, it might not be possible."
"I'm sure you can do it," said Stefano.
Can you find a suitable flight schedule?
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The answer
Let the cities be A, B, C, D, E, F and G. Decide that the first route will be ABC. As each pair of cities is on a route, another route must contain the cities AD. As each pair can only be on one route, the third city cannot be B or C, as that would conflict with ABC. It must be E, F or G. Decide it is G, making the second route ADG. So, for the remaining routes:
Cannot go with 3rd city must be Route
CD A, B, G Decide it is E CED
CG A, B, D, E F CFG
AF B, C, D, G E AEF
BG A, C, D, F E BEG
BD A, C, E, G F BDF
This is the only solution, although the letters can be permuted in any way.
In diagram form (right) it is called the Fano plane.
The winner is Tim Cole, London W4
