How does Santa visit and deliver all those toys to the good girls and boys around the world in just one night? With approximately 7 billion people on earth, that’s almost impossible to image. Where do we begin to even come to grips with quantifying this problem?

Fermi Problems are illustrative of how scientist and engineers are trained to think – to systematically break down a large problem into simplier descrete steps and logically solve them with clear, well-reasoned assumptions and approximations. Many high-level employers trying to fill cognitively demanding positions ask Fermi Problems in employee interviews to judge an applicant’s ability to think about complexity, think logically and think on their feet in real time (Google, Microsoft, Silicon Valley Startups, top global consulting firms, etc).

( The great physicist Enrico Fermi 1901-1954 )

Often known as “back of the napkin” type analysis, Fermi problems are a way to reduce a complex problem into simple, clearly defined steps that work together to answer the complex problem. A famous example is what is the probability that a breath you take today would contain any of the same matter that Julius Casear exhaled upon his death in 44 B.C. (The Last Breath of Cesear Fermi Problem)?

Assumptions and estimation should be derived from general knowledge or logical assumptions/derivative from such knowledge. Generally, one shouldn’t have to look up key statistics or use a calculator to flesh out an answer. To quote Wikipedia:

*“In physics or engineering education, a Fermi problem, Fermi quiz, Fermi question, or Fermi estimate is an estimation problem designed to teach dimensional analysis, approximation, and the importance of clearly identifying one’s assumptions. The solution of such a problem is usually a Back-of-the-envelope calculation. The estimation technique is named after physicist Enrico Fermi as he was known for his ability to make good approximate calculations with little or no actual data. Fermi problems typically involve making justified guesses about quantities and their variance or lower and upper bounds.”*

So let’s analyze just how Santa would be able to deliver all those toys on the night of December 24th. This is what I worked out with my boys on the car ride this morning.

(1) There are 7 billion people in the world, of which about 2 billion celebrate Christmas (Christian or culturally observant)

(2) A simple population pyramid of these 2 billion celebrants would have about 15% in the 1-12 year old target market for Santa’s good girls and boys (or 15% of 2 billion = 300 million)

(3) Most of the Europe, America and developed parts of NE Asian are suffering population decline (if not for immigration) so there are a lot childless adults and married couples. Assume an average family size of 4.0 (1.5 parents – many single parents + 2.5 children/couple where 2.1 children/couple is replacement level). So our 300 million children are clustered into 300 million/ 2.5 children/household = 120 million distinct households to visit.

(4) Assume that some households request that Santa deliver presents on some day other than the 24-25th of December and that some naughty children are underserving of any gifts. Let’s generously estimate this at 40 million to have a nice round number to work with in the next step. 120 million households with children – 40 million not delivered to on 24-25th = 80 million households

(5) Now say Santa can start work at 8pm in the evening on the International dateline and work 36 hours until 8am in the morning the next day on the International dateline. That would give Santa an approximate present delivery rate of 80 million household / 40 hours = 2 million households/hour

* As an aside, a real problem for Santa is the uneven population distribution. Although Santa could spend less or more than 36hrs/24 time zones per time zone depending upon the population density of each time zone, sparsely populated time zones like Islands of the Pacific Ocean and even rural areas will incur a lot of travel overhead. Fortunately for Santa, this aspect of his job has become significantly easier over the past +114yrs as the urban/rural populations shift toward urban (10/90 -> 90/10 in the USA).

Two million household visits/hr may sound like a lot, but it’s an order of magnitude less than our initial guess based simply on the planet’s population of 7 billion people. Luckily, Santa’s sleigh is the only object with mass that we know can travel near or at the speed of light and he has legions of labor extendering elves (although their jobs are in danger of automation by Google’s self-driving technology and recent acquisitions in the robot space).

When we got home I Googled for “Santa Fermi Problem” and sure enough, there were others who contemplated this problem. A detailed example of a deeper examination of the Santa question is given a physicist at Fermi National Labs. The money quote:

*“The distance Santa has to travel can be estimated from the following. First, while the surface area of Earth is about 10 ^{14} square meters, only about 30 percent of that is land mass, or about 0.3 x 10^{14} square meters. Second, we’ll assume, for simplicity’s sake, that the 800 million homes are equally distributed on this land mass. Dividing 0.3 x 10^{14} by 800 million gives 4 x 10^{4} square meters occupied by every household (about six football fields); the square root of that is the distance between households, about 200 meters. Multiply this by the 800 million households to get the distance Santa must travel on Christmas Eve to deliver all the children’s gifts: 160 million kilometers, farther than the distance from here to the sun.*

*Thanks to the rotation of the earth, Santa has more time than children might initially think. Standing on the International Date Line, moving from east to west and crossing different time zones, Santa has not just 10 hours to deliver his presents (from 8 p.m., when children go to bed, until 6 a.m., when they wake up), but an extra 24 hours— 34 hours in all.*

*Even so, Santa’s task is daunting.*

*Now, some have guessed that Santa accomplishes his task by traveling at a speed close to that of light—let’s say, 99.999999 percent of the speed of light. By traveling that fast, in fact, Santa can deliver all his presents in just 500 seconds or so, with plenty of time left over (the remainder of the 34 hours) to polish off the cookies the children have left him on their kitchen tables.”*

This is a much more rigourous examination of the question than I did with my son in the car because he solves for the estimated total path length Santa must travel, but most of our assumptions were in agreement. I did notice one major error in the assumptions made: Most humans on the planet do not celebrate Santa’s gift-giving which the author above assumes.

Merry Christmas and watch the color of Rudolph’s nose in the sky to estimate Santa’s speed on Christmas eve.