The Mathematics Of Santa Claus’ Present Delivery System

The Mathematics Of Santa Claus’ Present Delivery System
Science Gone Wild with William M Briggs
The Mathematics Of Santa Claus' Present Delivery System

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 scientific field of present dynamics.

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

“A math and weather wiz at NYC’s Cornell University helps crunch the numbers [about Santa]…” (A heavily compressed clip: if anybody has access to a better rendition, please let me know.)

Many mathematicians go to great lengths to prove, using various theorems and lemmas, that there is no way Santa could physically deliver all those 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 were only fast enough always ends disastrously. 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. And since those presents are delivered, 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 (and simplified) 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 helmet 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 looks at his watch and says to his assistant, “I’ve got 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 cause other puffs of air to divert from their course, 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 the subtleties of probability theory. The first branch describes how the present “moves” through world, from its place of origin to its spot under 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 momentum equation

Gift Probability Equation

Gift probability equation

Merry Christmas, and God bless us everyone!


  1. Lee Norris

    FANTASTIC read!!!!!

  2. Eric Stevens

    Super! I will now officially drop my adherence to the “Santa as a probability wave function that acts across the globe” in favour of this new insight! Of course its obvious now that you point it out – effective leverage of existing natural laws, much more impressive.

    Thank you for this!

  3. JH

    The odds of receiving presents from Santa decrease as I get older. No way… so not fair. I might as well be naughty and have fun then. I think \beta_2 is a negative number and therefore “-\beta_2” is positive.

  4. Pompous Git

    William/Mat/Briggs [delete whichever is inapplicable] This is wonderful! If laughter is the best medicine, then my recently broken leg should heal weeks earlier than predicted by the X-spurts.

    PS Thanks for the book. It arrived while I was in horsepiddle. I reciprocate your kind offer of a beer when you come to the Land of Under with free accomodation in the House of Steel and gourmet food from my kitchen.

  5. Joy

    Dear Santa’s helper,

    Really cute video, I used the biggest magnifier in Christendom, (15X)
    Imagine if your blog readers messed about like that.
    We wouldn’t dare, Mr. Briggs is too strict and we are all far too grown up, except Bernie, who never knowingly behaves himself.
    My nephew (6) will watch this, not that it will have any effect on his behaviour, as he is confident that Santa will come up with the goods. Either that or he believes Santa has a short memory. He loves numbers though. “Numbers are for boys, letters are for girls.” He informs me.

  6. meimie

    Santa gets everything out on time by freezing time. It says so in first Artemis Fowl book. lol

    Although Santa isn’t real. I figured that out when my Grandma used the same wrapping paper as Santa.

  7. Briggs


    Of course Santa exists. All my paychecks have gone through.

  8. Joy

    Don’t try and look innocent!

    Santa does the same with the paper as he does with the presents. It’s obvious. He…another thing worth bearing in mind is that there are only a limited number of papers available. I know this, I am an expert on wrapping parcels. This is one of the dangers of global markets.

  9. Oh come on Matt. There is nothing wrong with your looks.

  10. Bernie

    Those guys are going to have a hard time figuring out what caused the blip in their visits!!!

  11. Dang! I do need to pick up my new eyeglasses. The optometrist called, and they are in. . .

  12. Joy

    You and your friend were ok, I’d lose the baseball cap if I were you, but who was that painfully shy fellow with the sexy laugh? He was cute. He seemed to think he was a statistician.

  13. This is great work. I think the WSJ is calling for your talents:

    “Some smart mathematician should utilize divorce statistics to develop a software program that will predict recessions.”

    I look forward to the resulting post. Also, did I miss the Presidential wishcasting results?

  14. Briggs


    My only baseball cap says “New York Methodist Emergency Medicine”, so you know it wasn’t me.


    Haven’t done it yet! I working on it…

  15. Joy

    Oh, Dr. Briggs is a medical Doctor and has the hat to prove it;
    The other fellow, Matt, is the one with the baseball cap;
    William must be the tall one who is “statistician to Father Christmas”.
    Glad we cleared that up. It pays to know who’s who.

  16. Briggs


    No! I’m no MD. I just have the hat.

  17. Joy


    I’d keep quiet about the hat, you can be done for impersonating a Dr.

  18. The Weird US guy have a series of books, one for each state, that are well worth the money, and make great gifts. Without Weird Pennsylvania, I doubt I would have ever noticed that it does indeed look like William Penn is pissing of the top of the city county building in Philadelphia …

  19. Joy

    Isaac (just turned 7) says,
    “I can’t see it!…I can’t see it! That’s too small!” (he has 20-20 vision.)
    “That’s not Father Christmas!! Where’s Father Christmas?”
    When I explained a second time he was unimpressed, never having doubted that Father Christmas has it all under control. He’s too young really but I thought I’d see if my prediction was right.

  20. The kid told Santa he wanted “a football” but his DA Dad got him a “football helmet”. Typical Government employee!

  21. DAV

    Nice treatment of an enduring (dare I say “endearing”?) topic. I wonder though. With Santa’s propensity toward using chimneys as doorways I should think he is the original Soot Troll or at least the basis of the legend. Recently, though, someone seems to have stolen that that title.

    I notice that as I grow older, the number of homes with chimneys has declined. This may contribute to Santa’s continuing ability to distribute gifts as the population increases. Further study is indicated.

  22. Joy

    Merry Christmas everyone!

  23. AusieDan

    Hi Matt,
    Very good, althought I’m still puzzling over the formulae.
    But the story rang true to me.

    When my youger daughter was about four years old, she bailed me up and asked me straight out “was santa real?”
    Well – I couldn’t look here in the eye and lie, could I?
    (Double well – I just might have lied, if I had time to gather my thoughts together)
    So I just told her straight – the plain truth.

    “Well” she said, “where do all the presents come from?”
    “Mummy and I buy them for you and put them under the tree”.
    She looked at me and burst out laughing and went outside to tell the dog.

    She just knew my funny sense of humour and how stingy I was.
    She knew I never bought any of the toys she begged for during the year.
    So she concluded accurately, that Christmas would not be any different.

    She is over forty years of age now and to the best of my knowledge is still a true believer.

    There I hope some of you forgive me for my rudeness in my post to the previous item.
    Merry Christmas (there – I’ve got it right at last) – even if it’s now the 27th.

  24. Ron Number

    Next year do a Bayesian analysis of the problem, please. Because your frequentist analysis is a trifle Victorian, if you know what I mean.

  25. Mark Z

    What a fun read!

    From the “God bless us everyone” to “Santa Claus Gift Momentum Equation”, that was excellent humor.

    Merry Christmas (belated) to you too!

  26. JH

    After plugging the gift probability equation and using some simple algebra, as I have been nice and haven’t aged a bit, I shall get what I want (not an iPad) this year with a probabilistic of 0.99. The secret stuff really helps! ^_^

    I miss Joy and Bernie.

  27. JH

    I typed ‘probability’, not ‘probabilistic’!

  28. DAV

    A Blast from the Past!

    Of course there’s a Santa. Virginia’s research showed a positive result. Small poll sample though. (*sigh*)

    “The actual algorithms are, of course, secret and proprietary, so I can only give you a sketch here.”

    A non-disclosure agreement? Gosh even Santa is afraid of competition.

    What JH said, re Joy and Bernie.

  29. Briggs looks like a Happy Warrior in the clip–or the fellow in the Wodehouse novels who whispers, “I know your secret!”

    Just saw the movie Arthur Claus with son, seven years, the other day. We both liked it very much. Posits a sort of humongous Imperial Cruiser with stealth tech and few million elves aboard as the current sleigh. However, when the chips are down, Magic Dust is brought out to save the day.

  30. Sera

    I’m still waiting for the supplementary info on ‘Easter Bunny’.

    Merry Christmas!

  31. I thought it was obvious that the presents are delivered by a quantum superposition of Santas.

  32. Joy

    Merry Christmas,
    Thank you Jh and Dav.
    no Bernie?

  33. DNY

    All models of Santa Claus’s activities I’ve seen overlook the fact that Christmas presents in Russia, Serbia, Ukraine, Georgia, Egypt, Ethiopia, Eritrea, and to a goodly number of households in Alaska and the Patriarchate of Jerusalem and scattered household throughout the rest of the world are due on 7 January (by the civil calendar) rather than 25 December.


    (Orthodox monastic smiley)

  34. Joy

    Merry Christmas one and all!
    love from Joy X

  35. really mindblowing but i did not understand any thing ouy of it.

  36. My grandchildren know the answer to how Santa does it:

  37. Ken

    “SECRET STUFF” isn’t quite so secret since Harry Potter movies revealed some elements such as the tents (small outside, huge inside); “warp”-like speed travel & other high-speed non-broom flying that leaves some inky black [ectoplasmic-like?] dusty & quickly dissipating residue. Not to mention Cheech & Chong’s revelation some years back about the magic dust Santa uses to make himself small….

    Also, the bit about Chaos theory — the so-called “butterfly effect” — repeats an erroneous, ubiquitous, perspective. Chaos theory does NOT expect that a butterfly’s wings could alter wind or weather. The issue that imperfect analogy is intended to illustrate is that very tiny effects (e.g. the umpteenth decimal point) OF A LARGER ACTION have, over time & in the aggregate, potentially very large effects. I.E., Chaos theory would expect, for example, that the 25th decimal point value of a major effect, like incidence of solar energy input to a model, would lead to very divergent results relative to a seemingly acceptable round-off (e.g. the 2nd or 3rd decimal point as the subsequent magnitude is un-measurably tiny). This is because the cause of the effect being measured is necessarily & measurably significant to begin with. As for a butterfly’s (another “cause”) contribution, already inconsequential–but perhaps on par with those un-rounded-off decimals noted, the actual theory would expect the inconsequential contribution to remain inconsequential. This is a an example of the all-too-common problem of models & analogies being extrapolated beyond their relevance, and doing so unwittingly.

  38. It’s time for my standard explanation: We are dealing with a quantum superposition of Santas.

  39. John R T

    Merry Christmas, Matt
    The lady of the house decorated the tree, yesterday – while I napped: this may be my best Christmas, ever

  40. m .e. w

    This is like the medieval puzzle “How many angels can dance on the head of a pin”
    Innumerable angels ,since they are incorporeal
    Santa ( Saint Nicolas) would be outside of time, ( in eternity) and would therefore not be constrained by time. So he can deliver all gifts at appropriate times. St Nicholas Day would be one of those times for many children in Europe. Santa can also manipulate space thus allowing himself to enter rooms without a chimney. He used to leave mine on my bed, so that is proved.

  41. Gary in Erko

    Bah ! ! Humbug ! ! ! It’s silly to apply rational explanations to events that lie outside of rationality. Santa does it year after year all on his own. That’s all we need to know. Don’t attempt to limit the scope of his abilities with formulas and measurements.

  42. Gary in Erko

    Once upon a time when I was six years old, my three years elder brother wrote our notes for Santa’s delivery and we nailed the stockings to the fireplace mantlepiece. Santa, however, delivered our gifts to the tree of our next door neighbors becuse we’re Jewish so he doesn’t visit our homes. It took me many decades to realise the balancing my parents did, in allowing us to partake of the mainline culture while retaining elements of our own. It took a few extra years before I appreciated the assistance from our Greek Orthodox Christian neigbours in assisting with this. Sydney during the decade after world war 2 was a very healthy mixture of cultures before the word ‘multiculture’ was coined.

  43. All I want for Christmas is….
    1) a Republican nominee who is not Donald Trump
    2) Peace on Earth and Good Will to Men.

    May you all have a blessed/happy Christmas and a good New Year.

  44. This is the first plausible explanation I’ve seen for Christmas Eve present distribution. However, there must be errors in your Gift Momentum Equation, because it appears to be not dimensionally correct.

  45. In the equation is the time element missing, santa starts in new zealand and ends in hawaii after 24 hours.

  46. Steve E


    It seems this post has been visited by ghosts from Christmas past and present; can we expect a visit from the ghost of Christmas Yet to Come?

    Merry Christmas

  47. JH

    May kindness, love, and peace be yours in abundance.

    Today I hope my Berry Christmas pie and grispelle, made without using the two math equations above, tun out great!

  48. Sheri

    I’m more interested in who wrote the algorithm for the Santa tracker at Norad. Getting the image of Santa over the entire globe in 24 hours seems challenging. I was trying to discern if Norad’s Santa tracker and Google’s Santa tracker matched and it did not seem like they did. Could be confusing to kids if they find out Santa is in two places at one time.

  49. Trigger Warnings

    How does Santa do it?

    “Any sufficiently advanced technology is indistinguishable from magic.”
    — Arthur C Clarke

    I would explain how he does it, but I’m bound by a NDA.

  50. Scotian

    Thunderbirds are go!

  51. Steven Fraser

    Shared this with cousin Gerry Bowler, historian of all things Christmas.

  52. Santa causes the gifts to be delivered. Then, like DEATH, he personally delivers a few to maintain the symbolism, check up on the process, and maintain a link with us peculiar mortals.

  53. Thiago

    Since this post was first published, the mathematicians Dr. Hannah Fry & Dr. Thomas Oléron Evans have published “The Indisputable Existence of Santa Claus”, a volume entirely about the mathematics of various Christmas traditions. The topic of Santa’s delivery system, however, seems to have been left uncovered by these authors, presumably due to its thorough and solid treatment by Dr. Briggs.

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