The Mathematics of
Santa Claus


This page presents the creative efforts of a pair of MATHCOUNTS coaches from Washinton, D.C., and several members of their team. Guy Brandenburg and Joyce Higginbotham are the coaches, and the students include Harry Stein, Acecelle Ogbac, Nathaniel Beale, and Laura Flagg. Guy sent the material that appears below via email to "math-teach", a math discussion list from the MATH FORUM, where I found it. It is being used here with their permission.

The Email

Date: 24 Dec 98 From: Guy Brandenburg Subject: The mathematics of Santa Claus I thought this might amuse the assembled ears. The idea is not original, but we did our own research and computations (and made our own assumptions). It went over pretty well at the holiday assembly right before the winter break.

The Mathematics of Santa Claus

By The Alice Deal Junior High School Mathcounts Team (Washington, DC Public Schools) Narrated by Acecelle, Harry, Nathaniel, and Laura Acecelle: Happy holidays to you from the Deal Mathcounts Team! This year, inspired by something we saw on the Internet, we tried to figure out just how Santa Claus manages to bring all those presents. Harry: Not everybody in the US or in the world celebrates Christmas. We found in a recent World Almanac that there are 5.58 Billion people in the world, and of them, only 1.89 Billion are Christians. That leaves a lot of Moslems, Hindus, Buddhists, Jews, atheists, Sikhs, Jains, and members of other groups who mostly do not expect to get any presents from Santa at all. Nathaniel: We figured that Santa only gives presents to the children who are 15 and under. We found data in the almanac about what fraction of the population is younger than 15 years, for several countries (it varies). We estimated that about 1/4 of the Christmas-celebrating population would be kids who get presents. Laura: So that means we had about 473 million children that Santa needs to bring at least one present to. We didn't want Saint Nick to have to work too hard, so we figured that each GOOD child would get a 1-pound present of some sort, in a box that was about 10 cm by 10 cm by 20 cm.
Acecelle: The naughty kids would only get a lump of coal. (Show lump) That is much easier for Santa, since it only weighs about 4 ounces and is much smaller. We also decided that most kids --80% -- are nice, and only a small fraction -- 20% -- are naughty. Nathaniel: But this is still a lot of presents for Santa to bring: 80% of 473 Million is about 380 million presents! Harry: In fact, if each one weighs about 1 pound -- roughly 1/2 kilogram-- then that is about 190 Million kilograms of presents for the good kids, which works out to about 190,000 tons of goodies. Acecelle: The coal, as we said before, is much less: about 95 million lumps, each weighing 100 grams, is about 10,000 Tons of coal. That's nothing, if you compare it to the 190,000 tons of good stuff. Laura: We were wondering how big this pile of presents would be. You all know what RFK or Jack Kent Cooke stadium looks like? Well, they are roughly 120 yards long inside, about the same across at the top level, and maybe 60 yards tall. Nathaniel: Imagine an entire football stadium filled with boxes of presents -- to the top. That is about the size of the pile of presents and coal that Santa Claus has to lug around on Christmas eve. If he brings two presents to each kid, then that’s TWO football stadiums, and so on. Harry: One other thing: How many children are there per household that has kids? We realize that family sizes are not all the same, but to make the math easier, we assumed that there are, on the average, 2 children in each house that has kids. So that means that Santa must make about 236 million stops on Christmas eve. Laura: Santa also has another problem: What is the best path to use to get these presents as quickly as possible, without wasting time retracing his steps? It turns out that mathematicians have been trying to figure out this problem for about 100 years. The problem is so well-known that it even has a name: "The Travelling Salesman Problem". Acecelle: It turns out that nobody has yet solved the "Travelling Salesman Problem", not even with the fastest computers on earth, because there are so many different arrangements to check. But Santa --- he's really smart, so we decided that he has figured out the best possible path to take. Harry: We had no idea how far the average distance was going to be from one household to the next. After all, some people live in apartment buildings. Other people live spread out on farms. And of course, Santa needs to cross a whole bunch of oceans. We arbitrarily decided that there would be about 10 meters between each household. If you don’t like that, then you can do the math yourself! Laura: So that means that poor old Santa has to travel a total of about 2.36 million kilometers, or about 1.4 million miles on Christmas eve. Nathaniel: If Santa starts on one side of the planet just as it is getting dark on December 24th, and travels around the world in the correct direction, then before the sun rises on December 25th in the last possible location, he will have almost 40 hours to do his work. Acecelle: Since speed is equal to distance divided by time, we figure that these 1.4 million miles would be divided by 40 hours, which means that Santa only needs to go at an average speed of about 10 miles per second. That is about the distance from here to the Wilson Bridge, every second. Harry: Or look at it this way: 236 million stops, divided by 40 hours, means about 1600 households every second. Not every hour, but EVERY SECOND. Laura: We haven't figured out how he eats all those cookies, or drinks all that milk, or even gets into houses that don't have chimneys. That is a lot of stuff to eat and drink! Nathaniel: But remember that huge pile of gifts? He has to haul it with him, with a sleigh with a bunch of reindeer. It will take a lot of energy to haul all those gifts and that sleigh, and to make all those stops and starts. We looked at some physics formulas, and calculated that if there are 9 reindeer, then each one would have to generate 1.7 Billion horsepower – that’s Billion, with a B.

Acecelle: On the other hand, maybe these are just normal reindeer.  If
they are, and they  generate about 1 horsepower each, Santa would need
almost 2 Billion reindeer, which weigh probably 100 pounds each,
adding a lot of weight to his sled. Not only that, but if they are
hitched up to the sled 2 by 2, the line of reindeer would reach from
here to the Moon and back, TWENTY TIMES.  Let's hope these reindeer
are magical, and he won't quite need so many.

Nathaniel: One more problem, one that we haven't quite figured out: If
anything that big -- the size of RFK stadium -- is going that fast --
10 miles per second -- it is likely to (a) cause sonic booms loud
enough to knock down entire cities, and (b) to burn up in the
atmosphere because of friction, just like falling stars. Now that
would be a sight to see, if you survived the explosion!

Laura: So maybe we're all wrong. Maybe there is more than one Santa
Claus: he clones himself, or has Virtual Santas in every country, who
know the local language, too. But even if there are 100 Father
Christmases, those sonic booms would be noticeable; and what would the
air traffic controllers say?

Harry: Somebody suggested that maybe each household that celebrates
Christmas has its own individual Santa Claus. We thought about that
one, and said, 


Acecelle: Can't be – how could there be hundreds of millions of Santa
Clauses? That’s ridiculous. How could anybody believe something so

Harry: Maybe YOU can figure out how Santa does it all. But whatever
the answer is, however those presents get delivered, we just want to
say to all of you:


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