Dec 06 2007
Santa Claus Math
How does Santa Claus do it? How does he get all those presents to all those kids in just one night? Some people think that the old man still personally delivers all those presents just by himself. But there are too many kids now, so the traditional method has become impossible. So I was asked by the show Weird US to outline the modern mathematical ideas that Santa Claus actually uses.
There is no way Santa could physically deliver all the presents in just one night. This is because there are tens to hundreds of millions of children, and there is not enough time, energy, or space to complete the task in this short a time. A typical mathemtical analysis is this one by an engineer. His math and reasoning are flawless: any argument based on speed fails, which of course it must. However, those presents do get there, so Santa must be doing something. But what?
Have you see the movie Miracle on 34th Street? I mean the original, not any of the (unnecessary) remakes. There is a scene in the sanity trial of the old man who claims to be Santa in which the defense attorney calls to the stand the young son of the prosecutor. The prosecutor has previously argued that there is no Santa Claus. The defense attorney, John Payne, asks, (words to the effect), “Johnny, do you believe in Santa Claus?” The kid replies, “Sure I do.” Payne: “Why?” Kid: “Because my daddy told me (there was a Santa Claus).” Payne: “And your daddy is a very honest man, isn’t he? He wouldn’t lie?” Kid: “My daddy would never lie, would you daddy?” The kid comes off the stand and whispers to Santa that he’d like a football for Christmas.
Well, we all know what happens. The prosecutor conceeds the existence of Santa and the court eventually decides that the old man in the dock is the one and only Santa Claus. But the key scene sneaks by unless you’re paying close attention. It’s when the case is over and people are noisly exiting the courtroom. We see the prosecutor suddenly realize that he’s got to run. He says to himself, “I’ve got to hurry if I’m going to get that football helmut!”
To be obvious: the kids asks Santa for the helmut, but it is the father who brings it. Do you see? Santa manipulated the events so that the kid got what he wanted for Christmas — Santa was responsible for the present — but Santa did not actually, physically have to bring the present. Here’s how.
Have you heard of chaos theory? This is the mathematical theory of how things move when they are under complex or unidentifiable forces. The common description is that a butterfly flapping its wings in Brazil eventually causes a snow storm to develop in Cleveland two weeks later. How? Well, the tiny puffs of air forced from the flapping of the butterfly’s wings grow and magnify, eventually pushing their way north and building into a storm. Point is: a small cause can grow into a large event later. You can imagine that the mathematics to track such events are difficult.
Now, Santa doesn’t do this math himself. His specialty is in toy making, not differential calculus, so Santa employs a group of consultants to help with the complicated computer code that is necessary to bring about the massive toy movement on Christmas Eve. I am one of those consultants and so can give some hint about how things work. The actual algorithms are, of course, secret and proprietary, so I can only give you a sketch here.
Santa’s sleigh ride is largely ceremonial at this point, though he does go out and personally deliver some presents. He does this in cases where the math indicates that certain children are unlikely to get exactly what they want. And this is because the methods that we use are not perfect: Santa and his elves can only “flap thier wings” in so many places and in so many ways.
There are two main branches of math: the physics of chaos theory, and the statistics of probability theory. The first describes how the present “moves” through world, from its place or origin to the proper Christmas tree. This is describes in the “Santa Claus Gift Momentum Equation”, shown below. The bold “V_gift” describes, in three dimensions, the actual physical location of the present. The other parameters are the forces which govern that movement.
Now, the parameters in the momentum equation are decided by the probability equation, given next. The “p” in the equation is a probability, which should give you some hint that these methods are not perfect. Pay attention to the “I(Nice)” function. That is the “naughty or nice” indicator. Yes, Santa still keeps track of these things, so be careful! There is also a lot of “secret stuff” in these equations. But if you are too curious and just need to know, the best thing is to study physics or math and then someday, if you get good at it, Santa may ask you to help him with Christmas.
| Santa Claus Gift Momentum Equation |
|---|
|
|
| Gift Probability Equation |
|---|
|
 |
