William M. Briggs

UPDATE: The show Weird US—It’s a Wonderful Time to Be Weird on the History Channel will be airing episodes which feature this post on Tuesday December 23 11:00 AM and Tuesday December 23 05:00 PM. See this page for the schedule.

“A math and weather wiz at NYC’s Cornell University helps crunch the numbers [about Santa]…”

It’s the time of year when people begin asking the very pertinent question: 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 hand delivers each and every toy—with the enthusiastic help of Dasher and others, of course. That used to be the case, a very long time ago, but there are too many kids in the world now, and the traditional sleigh-bearing method has become obsolete and even impossible.

About a century ago, Santa saw what was coming and began to devise new present-delivery techniques. Naturally, Santa, being the world’s greatest manager, knew that he couldn’t figure out how to do everything all by himself, so he hired outside consultants. I am one of these (not one of the first, of course; I came on only in the last ten years). My contributions are in the field of present dynamics.

A couple of years ago, I was asked by the show Weird US to outline the modern mathematical ideas that Santa Claus now employs. The episode in which I appear (near the end) is entitled “It’s a Wonderful Time to Be Weird.”

Many mathematicians go to great lengths to prove, using various theorems and lemmas, that there is no way Santa could physically deliver all the presents in just one night. Arguments begin by noting that 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 analysis is this one, by an engineer. His math and reasoning are flawless. In fact, any argument which attempts to show that Santa could do his job if he was only fast enough always ends disastrously. For example, Santa would have to travel so fast that the reindeer would burn up like meteors entering the atmosphere. However, these mathematical results, while true, are answering the wrong question. Those presents do get there, so Santa must be doing something else. 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 concedes 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 noisily 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 helmet!”

To be obvious: the kids asks Santa for the helmet, 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 it’s done.

Have you heard of chaos theory? This is the mathematical theory of how things move when they are under complex or unidentifiable forces. A common example: a butterfly flaps its wings in Brazil, and eventually a snow storm develops in Cleveland two weeks later. How? Well, the tiny puffs of air forced from the flapping of the butterfly’s wings change other puffs of air to divert, which in turn cause still others to change their course, and so on. The effect grows and magnifies so that the path and dynamics of a future storm is changed. Point is: a minuscule cause can grow into a macroscopic 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 have been given permission to 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. This is because the methods that we use are not perfect: Santa and his elves can only “flap their wings” in so many places and in so many ways.

There are two main branches of present dynamics mathematics: the physics of chaos theory, and that of probability theory. The first branch describes how the present “moves” through world, from its place or origin to the proper Christmas tree. This is described in the “Santa Claus Gift Momentum Equation”, shown below. The bold “V_gift” describes, in three dimensions, the actual physical location of the present at any moment in time. The parameters of that equations 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! You can see that the coefficient on Age is negative, meaning that as you get older, you are less likely to get the present you want. There is also a lot of “secret stuff” in these equations that I can’t show you. 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