Christopher Rodrigues, chairman of VisitBritain, was stumped. This rare occurrence came about at an event where he found himself sitting next to the boss of China Southern Airlines – one of Asia's biggest, carrying 80 million passengers a year, and dwarfing almost all European carriers.
The subject turned to how the two might work together to entice more Chinese visitors to the UK. Mr Rodrigues reports that his companion expressed frustration at the shortage of slots at Heathrow: "I can't land in Britain anywhere I want to. If you can fix that, I'll send more tourists."
As airports across Britain, from Gatwick to Glasgow, implore new airlines to fill the yawning gaps on their departure boards, Heathrow is a different country. Passengers pay a premium to fly to and from Europe's busiest airport. Accordingly, "LHR" is the one UK gateway that Mr China Southern, and many other airline bosses, want to serve. The problem: Heathrow's take-off and landing slots are the the most precious commodities in aviation.
At present no British airport has links to China Southern's deep-south hub of Guangzhou, population 12 million. However much Gatwick or Manchester might move heaven, earth and landing fees to entice the airline to launch flights from China's third city, the only way is Heathrow.
But the airline knows that the airport is effectively shut to new arrivals, or at least to big airlines keen to run a meaningful schedule at reasonable times.
Sneaking on to the departure screens, amid all the jumbo jets bound for Asia and the Americas, are a handful of small towns whose very presence seems an aberration. Finding themselves on the world's elitest radar screen are the 180,000 residents of St John's in Canada (the same population as Preston); the 35,000 citizens of Beja in rural Portugal; and Kiruna's 20,000 shivering souls in the far north of Swedish Lapland.
Why are small jets flying obscure routes when they could be handling 500-seater planes bound for China? Because they serve profitable niche markets. Air Canada charges £775 return from Heathrow to St John's next month – about the same as the fare to fly more than twice as far, to Vancouver. Passengers to St John's will pay a fat premium to sit in a thin plane.
Beja is even more of a minority sport. Sunvil, the independent tour operator, reckons some folk will pay handsomely for direct flights to Portugal's beautiful Alentejo region. But not too many. So it dispatches only one 49-seat jet each week. By choosing the anti-social departure time of 6am on a Sunday morning, with the return hop arriving at lunchtime the same day, Sunvil has secured access to and from Heathrow from the airport's bin-ends, that small pile labelled "the 0.5 per cent of slots at odd times that no one wants".
And Kiruna? Well, the Scandinavian airline, SAS, has had a 10.35am slot for a flight from Heathrow to Stockholm since the dawn of time. It's the return leg of the (very profitable) first flight of the day from the Swedish capital to London, and is often half-empty. A sub-optimal slot, indeed.
Another enterprising tour operator, Discover the World, persuaded the airline occasionally to redirect the 10.35am departure to fly non-stop to the Arctic instead of Stockholm, giving British travellers access to the Ice Hotel and Northern Lights. On 10 days between December and March, you can slide off on a reindeer-hauled safari just four hours after take-off from Heathrow – so long as you have booked your slot on the sleigh.
Birmingham, gateway to ... Turkmenistan
For Heathrow's rivals, the airline schedules must make frustrating reading. Tomorrow, for example, 16 wide-bodied planes line up at LHR to fly you to New York's JFK airport. The rest of the UK can marshal only two departures to America's main gateway, both of them narrow-bodied jets from Manchester. But Heathrow doesn't get everything its way.
Most of the popular Mediterranean holiday spots, from Alicante to Zante, are unavailable from Heathrow. If you are dreaming of Ashkabad in Turkmenistan (left), Birmingham boasts four flights a week compared with Heathrow's two.
Heathrow doesn't offer the widest choice of destinations, either, with a mere 193 this summer. Manchester and Gatwick boast more diverse departure boards, each showing over 200 destinations.