## Fun With Math-Volume 2

The idea behind this post was hatched in my 2nd period Algebra II class while teaching 3 years ago when a student of mine filled out bracket after bracket for the upcoming NCAA Men’s Basketball Tournament (instead of doing his homework, of course).

I asked him if he was trying to try to fill out the perfect bracket by filling out every possible bracket. Then I asked myself (and the class) how many brackets would it take to ensure you had the right one. Here is the math:

(All of this ignores the completely idiotic “Opening Round” game and instead goes with the appropriate 64 teams bracket)

A 64 team tournament features 63 individual games, as the number of games played in a balanced (power of 2) tournament is the number of teams – 1.

In a 4 team tournament, there are 3 games and 8 total outcomes if no placement games are played.

2^Number of Games

Therefore, a 64 team bracket would yield 2^63 outcomes. This is a staggeringly high number, being roughly 9.223 x 10^18 outcomes, or for the rest of this post, separate sheets of 8.5 x 11 inch (letter-sized) paper.

How big is 9.223 x 10^18? Big enough to have its own name: We’ll dub it a Bracketplex.

That’s 1.8446 x 10^16 reams of paper, as 1 ream of paper = 500 sheets. Allowing for 2 inches per ream, stacked one on top of another, this would represent a pile 5.8226 x 10^11 *miles* tall.

Given that the distance from Earth to the Sun is roughly 92,000,000 miles (1 Astronomical Unit or AU), the stack of brackets would therefore be over 6,260 AU long. It is also a given that light travels at the approximate speed of 187,000 miles / second. For light to travel this staggering distance of 6,260 AU, it would take 54.06 days (.148 light years) to get from one end to the other.

The numbers are even more staggering if the sheets are laid end to end. The length of a Bracketplex jumps to 1.6012 x 10^15 miles, or enough for 17,217,369 AU (remember that’s a trip to the Sun from Earth), or enough to make 3,202,430,556 round trips to the Moon.

The Earth’s surface area (the amount of area it would take to wrap the planet like a gift) is roughly 7.906 x 10^17 square inches. A simple area calculation gives us 93.5 square inches for 1 sheet of paper (or one bracket). Covering the entire planet with a Bracketplex equates to laying down 1 sheet 1,090 times, wallpapering Earth with over 2 reams worth of paper, adding up to about 4.36 inches deep. This includes the oceans and lakes of the world.

Speaking of the world, what kind of production will the quest for the perfect bracket require to complete? The selection committee does not release the final bracket until Sunday evening at approximately 6:00 pm Central. The first game tips on Thursday morning at 11:00 am, allowing about 87 hours to complete our Bracketplex.

Given that there are about 6 billion people on the planet, it would take a gargantuan effort from every person, no matter how old or young, no matter what state of health or basketball knowledge. Each person on Earth would need to complete:

- 17,668,582 brackets per hour
- 294,476 brackets per minute
- 4,907 brackets per second

Keep in mind that while the Worldwide Printing Press is going, each bracket from each person has to be **UNIQUE**…each bracket filled out has to be distinctly different from every other sheet completed.

Sticking to the United States, if we were to make an 8-lane interstate completely out of a Bracketplex what would it look like? First, the amount of highway miles in the US Highway System currently sits at 4 million miles.

If we were to pave the US highways with brackets it would constitute a road 240 feet wide, stretching all 4,000,000 miles, and be a whopping 394 feet thick. That’s thicker than Rick Barry.

I performed a search on the internet about how many sheets of paper an average tree could yield. The results varied greatly, from 9,000 sheets to 80,500 sheets. Let’s go with the upper end to hopefully depress the number of trees we will have to slay.

It would take 1.1456 x 10^14 trees to fill out a Bracketplex. That amounts to 361,484 trees per square mile of Earth, regardless of water, mountains, ice sheets, or people and animals. Limiting it to just land, the figure rockets to 1,246,5000 trees per square mile. And they all have to cut down and processed this year…then regrown before next March. Madness indeed!

All of this, of course, has to be repeated next year. Let’s all do our part and plant those trees and recruit your neighbor and his kids to fill out 2010’s Bracketplex to make sure we get that $1,000,000 prize offered for the Perfect Bracket.

*~Greg*