December 2000

# The Mathematics of Christmas

I guess it was an early sign that I was heading for a career in mathematics that, when I was a young child, the run-up to Christmas always presented me with a numerical puzzle. How could Santa Claus possibly visit all children at midnight on the same night? I never did get a satisfactory answer from my parents, whose stock response was "No one knows; he just does."

These days, the adult me can address the question in a mathematically more sophisticated way. Just how big is the task facing Santa on Christmas Eve?

Let's assume that Santa only visits those who are children in the eyes of the law, that is, those under the age of 18. There are roughly 2 billion such individuals in the world. However, Santa started his annual activities long before diversity and equal opportunity became issues, and as a result he doesn't handle Muslim, Hindu, Jewish and Buddhist children. That reduces his workload significantly to a mere 15% of the total, namely 378 million. However, the crucial figure is not the number of children but the number of homes Santa has to visit. According to the most recent census data, the average size of a family in the world is 3.5 children per household. Thus, Santa has to visit 108,000,000 individual homes. (Of course, as everyone knows, Santa only visits good children, but we can surely assume that, on an average, at least one child of the 3.5 in each home meets that criterion.)

That's quite a challenge. However, by traveling east to west, Santa can take advantage of the different time zones, and that gives him 24 hours. Santa can complete the job if he averages 1250 household visits per second. In other words, for each Christian household with at least one good child, Santa has 1/1250th of a second to park his sleigh, dismount, slide down the chimney, fill the stockings, distribute the remaining presents under the tree, consume the cookies and milk that have been left out for him, climb back up the chimney, get back onto the sleigh, and move on to the next house. To keep the math simple, let's assume that these 108 million stops are evenly distributed around the earth. That means Santa is faced with a mean distance between households of around 0.75 miles, and the total distance Santa must travel is just over 75 million miles. Hence Santa's sleigh must be moving at 650 miles per second -- 3,000 times the speed of sound. A typical reindeer can run at most 15 miles per hour. That's quite a feat Santa performs each year.

What happens when we take into account the payload on the sleigh? Assuming that the average weight of presents Santa delivers to each child is 2 pounds, the sleigh is carrying 321,300 tons -- and that's not counting Santa himself, who, judging by all those familiar pictures, is no lightweight. On land, a reindeer can pull no more than 300 pounds. Of course, Santa's reindeer can fly. (True, no known species of reindeer can fly. However, biologists estimate that there are some 300,000 species of living organisms yet to be classified, and while most of these are insects and germs, we cannot rule out flying reindeer.) Now, there is a dearth of reliable data on flying reindeer, but let's assume that a good specimen can pull ten times as much as a normal reindeer. This means that Santa needs 214,200 reindeer. Thus, the total weight of this airborne transportation system is in excess of 350,000 tons, which is roughly four times the weight of the Queen Elizabeth.

Now, 350,000 tons traveling at 650 miles per second creates enormous air resistance, and this will heat the reindeer up in the same fashion as a spacecraft re-entering the earth's atmosphere. The two reindeer in the lead pair will each absorb some 14.3 quintillion joules of energy per second. In the absence of a NASA-designed heat shield, this will cause them to burst into flames spontaneously, exposing the pair behind them. The result will be a rapid series of deafening sonic booms, as the entire reindeer team is vaporized within 4.26 thousandths of a second. Meanwhile, Santa himself will be subjected to centrifugal forces 17,500 times greater than gravity. That should do wonders for his waistline.

Christmas is indeed a magical time.

The above column is adapted from a longer piece that has been circulating on the web over the past few years. I have no idea where it originated. If you've seen it before, my apologies. I thought it was worth giving a new outing.
Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.