December 18, 2016

Foto: Santa Clause and his trip around the world (Photo: CC0)

In order to deliver Christmas presents as quickly as possible, Santa Claus has to find the shortest trip around the world.

In mathematics, this task is called the Asymmetric Traveling Salesman Problem (ATSP). The calculation of the the flight time between two waypoints can be considered as a special type of Shortest Path Problem, which is called Flight Route Planning Problem. In order to calculate optimal flight routes, airlines today use mathemathematical planning systems. The shortest flight connections are calculated in this way, taking into consideration flight times, fuel consumption and weather.

For Santa Claus we consider a simplified version with constant speed (900 km/h) and altitude (10 km) and without rules of the air. The route is calculated on the basis of 1305 destinations around the world, starting at the North Pole. The ground speed, that is the speed relative to the earth's surface, is highly dependent on air pressure and temperature. With tail wind, the ground speed is higher, with head wind, it is lower and Santa Claus has to counteract in order to balance cross winds. To save time, Santa Claus does not land on the way but throws the presents off from the air for the regional Christmas elves to distribute. Since he has a bottomlos gift sack, the load of gifts for 1 billion children has not been included in the calculation. In order to make the round trip, Santa has to stop the time. Thus, the weather is constantly the same as on Christmas Day 2015 23:00 UTC. Under these conditions, a connection with minimal fuel consumption (or reindeer food) also has the shortest flight time.

If you want to retrace the tour, you can download Santa's route here as KML file and open it, e.g., in Google Earth (or your own navigation system).

The presumably optimal tour leads from the North Pole via Europe, the Middle East, to Russia and from there to Asia and back to Africa. It then goes on to North and South America with a stopover in Hawaii, to Papua Neuguinea and Australia. After a couple of landings in Asia, the Arabian Peninsula and Europe, Santa can head towards his well-deserved end of a long working day at the North Pole. The tour was computed using the TSP solver Concorde (after symmetrization), the flight times/distances with the flight path planner LIDO/VOLAR.

Crossovers, loops and what may seem to be detours on the route are no mistakes! A longer flight route can be faster and less fuel-consuming than the seemingly best direct connection on the great circle. In addition, Santa has, as a role model for the kids, to fulfil the requirements to the air route network. This global network has about 300,000 routes on each of the 30 permissible altitudes. The round trip shown is 365,174 km long, which is slightly more than 9 times the circumference of the earth, and the fuel consumption (when using an Airbus A380 with an average elf crew) is 5656 t. Thus, if Santa chose to go by plane, this would be a huge amount of fuel. Now we know why he usually prefers to go by his far more economical reindeer sleigh.

The flight time is approximately 406 hours or nearly 17 days. Unfortunately, this route crosses the International Date Line on its way from West to East, so that Santa loses yet another day. Fortunately, however, he is able to stop the time, so that he completes the tour successfully every year. And he surely will do so this year  – Merry Christmas!

When trying to find an answer to this question, we were supported by Forschungszentrum Matheon, researchers Marco Blanco, Prof. Dr. Ralf Borndörfer, Nam Dung Hoang, Pedro Maristany and Adam Schienle at Zuse Institute Berlin as well as Anton Kaier and Swen Schlobach at Lufthansa Systems.