Christmas Math 101

'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.

Sponsor: Want to save more, invest for the future, but don't have time to be a full-on investor? Betterment.com helps you build a customized, low-cost portfolio that suits your goals. Sign up at http://ift.tt/2hR2phJ and receive a $25 bonus when you make a deposit of $250 or more.

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?

---

How Fast Do Santa's Reindeer Fly?

This time crunch leads to another great Christmas math 101 question: How fast do Santa's reindeer need to fly? While this seems like a tough question to answer, remember that we're really just looking to come up with a quick and dirty estimate—so we shouldn't be afraid to use our trusty mathematical intuition and to make reasonable simplifying assumptions.

That's about 1,250 miles per second. Pretty zippy! In fact, it's several thousand times the speed of sound! Those are some mighty impressive reindeer.

For example, to answer this question we need to come up with a rough estimate of the average distance between the 92 million houses that Santa visits. To do so, let's simplify the situation and make the assumption that each of these 92 million houses are spaced equally across the entire Earth. Obviously this isn't actually true since there are oceans, mountain ranges, and lots of other obstacles in the way. But it is a reasonable first approximation for us to begin with.

According to WolframAlpha, the Earth's surface area is about 200 million square miles. Each of the 92 million houses that Santa delivers to therefore occupies a patch of Earth that's 200,000,000 square miles / 92,000,000 or approximately 2.2 square miles in size. In other words, if you divide the Earth up into a bunch of 2.2 square mile patches, each patch would contain an average of 1 house that Santa delivers to. If we take the square root of 2.2, we find that the length of the side of this patch—which is also the rough distance between houses—is approximately 1.5 miles.

So how fast do Santa's reindeer have to fly? Well, if each of those 92 million houses are an average of 1.5 miles apart, Santa has to travel 92 million x 1.5 miles = 138 million miles! As we've seen, if Santa uses Earth's rotation to his advantage, he has up to 31 hours of darkness for his journey. So if he and his reindeer travel 138 million miles over the course of these 31 hours, they'll be moving at a very brisk average speed of 138 million miles / 31 hours = 4.5 million miles per hour.

That's about 1,250 miles per second. Pretty zippy! In fact, it's several thousand times the speed of sound! Those are some mighty impressive reindeer indeed.

Wrap Up

Okay, that's all the math we have time for today.

Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Happy holidays and thanks for reading, math fans!

Betterment LLC is an SEC Registered Investment Advisor. Brokerage services are offered by Betterment Securities, an SEC registered broker-dealer and member FINRA/SIPC. Investments are not FDIC Insured. No Bank Guarantee. May Lose Value. Investing in securities involves risks, and there is always the potential of losing money when you invest in securities. Before investing, consider your investment objectives and Betterment's charges and expenses. Not an offer, solicitation of an offer, or advice to buy or sell securities in jurisdictions where Betterment and Betterment Securities are not registered.

Calculator gift image courtesy of Shutterstock.