# Fun Christmas Math

Did you ever wonder how many total gifts someone receives as a result of the song the 12 Days of Christmas? Or what individual gift was the most popular?

What about Santa Claus and his delivery schedule? How many houses does he visit? How fast does he have to be traveling? How heavy is his sleigh?

Well, you can find the answer to all those questions right here.

With regards to the 12 days of Christmas, a Dr. Wilson at Sonoma University has put together the following details. First, a quick summary of the gifts:

1. A partridge in a pear tree,
2. Two turtle doves,
3. Three french hens,
4. Four calling birds,
5. Five gold rings,
6. Six geese a-laying
7. Seven swans a-swimming,
8. Eight maids a-milking,
10. Ten lords a-leaping,
11. Eleven pipers piping,
12. Twelve drummers drumming.

We can count the total number of each gift.

 She gets 1 partridge in a pear tree on each of the 12 days. 1 x 12 = 12 She gets 2 turtle doves on the last 11 days. 2 x 11 = 22 She gets 3 french hens on the last 10 days. 3 x 10 = 30 She gets 4 calling birds on the last 9 days. 4 x 9 = 36 She gets 5 gold rings on the last 8 days. 5 x 8 = 40 She gets 6 geese a-laying on the last 7 days. 6 x 7 = 42 She gets 7 swans a-swimming on the last 6 days. 7 x 6 = 42 She gets 8 maids a-milking on the last 5 days. 8 x 5 = 40 She gets 9 ladies dancing on the last 4 days. 9 x 4 = 36 She gets 10 lords a-leaping on the last 3 days. 10 x 3 = 30 She gets 11 pipers piping on the last 2 days. 11 x 2 = 22 She gets 12 drummers drumming on the last day. 12 x 1 = 12 Total 364

So not a bad total for just 12 days – 364 gifts. We can also see from the table above that there was a tie for the most popular gifts; the recipient received 42 swans a swimming and 42 geese a laying. I hope the person getting all these gifts has a big enough place to keep all of it.

Now on to Santa Claus. I found some info from an old post written by Keith Devlin back in 2000 and is now archived at the Mathematical Association of America website.

Let’s assume that Santa only visits those who are children in the eyes of the law, that is, those under the age of 18. There are roughly 2 billion such individuals in the world. However, Santa started his annual activities long before diversity and equal opportunity became issues, and as a result he doesn’t handle Muslim, Hindu, Jewish and Buddhist children. That reduces his workload significantly to a mere 15% of the total, namely 378 million. However, the crucial figure is not the number of children but the number of homes Santa has to visit. According to the most recent census data, the average size of a family in the world is 3.5 children per household. Thus, Santa has to visit 108,000,000 individual homes. This assumes that there is at least one good kid in each household.

Traveling east to west, Santa can take advantage of the different time zones, and that gives him 24 hours. Santa can complete the job if he averages 1250 household visits per second. In other words, for each Christian household with at least one good child, Santa has 1/1250th of a second to park his sleigh, dismount, slide down the chimney, fill the stockings, distribute the remaining presents under the tree, consume the cookies and milk that have been left out for him, climb back up the chimney, get back onto the sleigh, and move on to the next house. To keep the math simple, let’s assume that these 108 million stops are evenly distributed around the earth. That means Santa is faced with a mean distance between households of around 0.75 miles, and the total distance Santa must travel is just over 75 million miles. Hence Santa’s sleigh must be moving at 650 miles per second — 3,000 times the speed of sound. A typical reindeer can run at most 15 miles per hour.

Assuming that the average weight of presents Santa delivers to each child is 2 pounds, the sleigh is carrying 321,300 tons. On land, a reindeer can pull no more than 300 pounds. Now, there is a dearth of reliable data on flying reindeer, but let’s assume that a good specimen can pull ten times as much as a normal reindeer. This means that Santa needs 214,200 reindeer. Thus, the total weight of this airborne transportation system is in excess of 350,000 tons, which is roughly four times the weight of the Queen Elizabeth.

Here comes the frightening part.

350,000 tons traveling at 650 miles per second creates enormous air resistance, and this will heat the reindeer up in the same fashion as a spacecraft re-entering the earth’s atmosphere. The two reindeer in the lead pair will each absorb some 14.3 quintillion joules of energy per second. In the absence of a NASA-designed heat shield, this will cause them to burst into flames spontaneously, exposing the pair behind them. The result will be a rapid series of deafening sonic booms, as the entire reindeer team is vaporized within 4.26 thousandths of a second. Meanwhile, Santa himself will be subjected to centrifugal forces 17,500 times greater than gravity.

Since such an approach is not sustainable, an alternative scenario has been suggested:

His reindeer — don’t actually pull a sleigh loaded down with toys. Instead, each house becomes Santa’s workshop as he utilizes a nano-toymaker to fabricate toys inside the children’s homes.

No matter how Santa and his reindeer do it, there is clearly magic at work, and that’s part of the beauty of Christmas – the mystery of it all.

As a famous scientist once said, not everything that counts be counted…

*image from Quick and Dirty Tips