Breaking Down the Mathematics behind March Madness

Jacob Shiohira
Aug 25, 2017 · 3 min read

Have you ever wondered why there is essentially a zero percent chance that anyone ever creates the perfect March Madness bracket? Sure — that’s what everyone always says, but I think it’s time we take a look into it.

The annual tournament starts in March (how fitting!) with a total of 64 collegiate teams in the first round. Not long after the first tipoff are fans upset by busted brackets, which can be quite upsetting considering that they missed out on a 1-in-2 chance of guessing the first game correctly. Oh how far that is from the 1-in-9,223,372,036,854,770,000 chance of filling out a completely perfect bracket. Yeah, that’s right… With 1 significant figure, we’d be looking at a 9 followed by 18 zero’s. It’s really hard to even comprehend how large that number is because… well, it’s just HUGE.

How does someone even go about arriving at that number for the total possible bracket combinations?

Glad you asked! It turns out, this insanely large number of unique bracket combinations can be calculated quite easily after we understand a few basic ideas in combinatorics — bet you didn’t think you’d be doing combinatorics today, did you? So, let’s take a few (or a lot) steps back from the initial 64 teams to just 2 teams. How many possible outcomes are there when 2 teams play? EASY, you say! And that’s true — when there are 2 teams, there is 1 game that will be played. Thus, there are 2¹ possible combinations,

Figure: Possible outcomes for 2 teams

Then, we can extend this methodology for 4 teams. When there are 4 teams, there are 3 games that will be played. Thus, there are 2³ possible combinations,

Figure: Possible outcomes for 4 teams

We can see from the figure that the possible combinations for 4 teams is constructed from the possible combinations figure for 2 teams. There are 2 possible outcomes for the first level of games between t1, t2 and t3, t4 — making 4 total combinations so far. Then, we see that from all the combinations of the second level game, we can have 4 possible winners — adding 4 more to the total combinations. Well, 4+4=2³=8. Way cool, huh?

It turns out that this methodology can be scaled for n teams such that the total combinations of outcomes for n teams is 2^(n-1). That means that when March Madness starts round one with 64 teams, the total number of outcomes (or bracket combinations) is equivalent to 2⁶³ = 9.223e10¹⁸ (aka that really large number we saw at the beginning. Now, you’ve done a little math and hopefully understand just how improbabile a perfect bracket is.

Cheers!

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Jacob Shiohira

Written by

Computer Science & Mathematics student at University of Nebraska-Lincoln. Previously @ Amazon & Microsoft. More about me at www.jacobshiohira.com.

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