Christmas Math 101

How long does Santa have to hang out when he's dropping presents off at your house? And how fast do his reindeer fly? Keep on reading to find out about the incredible math behind Santa's magical night!

Jason Marshall, PhD
4-minute read
Episode #179

Christmas Math 'Twas the week of Christmas, and all across the land
Math fans were thinking about Santa and his merry band;
How fast must they fly? How quickly must they go?
To make it to my house before sunlight hits snow.
Math Dude listeners were, of course, well aware
That Santa used math to help him prepare;
So they set out to use that very same math,
To try and determine his reindeer's flight path;
And once they learned how he does take that flight
They said "Happy Christmas to all, and to all a good night."

True story.

Well, okay—maybe it's not entirely true. Perhaps you've never ran into the streets to proclaim "Happy Christmas to all," but I bet you've wondered about the logistics of Santa's harrowing flight. And for good reason because the math behind that magical sleigh ride is pretty impressive. And it makes for a fun workout to help strengthen your mental math muscles. So, without further ado, let's take a look at Christmas Math 101.


How Long Does Santa Have?

The first question math fans might wonder is exactly how long does Santa have for his incredible journey? All night, right? Well, yes—but as we'll see, that's actually more time than you might think. And an even more interesting question is: How much time does Santa have to drop off your presents and eat those cookies you left him?

According to the always fascinating website snopes.com, the now defunct magazine Spy—or perhaps others before—published a fun look at the math involved here in its February 1991 issue. Here's the logic from that Spy magazine article:

  • There are 2 billion children in the world—we'll assume these are the only people Santa is delivering to.
  • Lucky for Santa, only about 380 million of these kids celebrate the Christmas holiday.
  • Assuming an average of 3.5 children per household worldwide, there are 380 million / 3.5 = 108.6 million houses eligible for delivery.
  • Assuming that 15% of houses have only naughty kids, Santa catches a break and has to deliver to only 0.85 x 108.6 million = 92 million houses!

Seems like a tough task, right? Well, this is where Santa gets some good news…and some bad news. As written in Spy:

"Fortunately, Santa has 31 hours of Christmas Eve darkness to visit all these homes if he travels from east to west, thanks to the rotation of the earth. Unfortunately, this still works out to 822.6 visits per second. So, for each household with good children, Santa has just over a thousandth of a second."

In other words, Santa doesn't have much time for sightseeing!

Next up, what about those speedy reindeer?


About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.