Bottle, Burger, Beer, Hot Dog Viral Puzzle – Mind Your Decisions

Thanks to Phil for the suggestion! As viral puzzles became popular, mathematicians joined the game too. Here’s a fun puzzle that has been widely shared.

Can you figure it out?

As usual, watch the video for a solution.

Bottle Burger Beer Hot Dog Viral Puzzle

Or keep reading.
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Answer To Bottle Burger Beer Hot Dog Viral Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

First we solve the three linear equations.

(bottle) + (bottle) + (bottle) = 30
(bottle) + (burger) + (burger) = 20
(burger) + (2 beers) + (2 beers) = 9

(bottle) + (bottle) + (bottle) = 30
3(bottle) = 30
(bottle) = 10

(bottle) + (burger) + (burger) = 20
10 + 2(burger) = 20
2(burger) = 10
(burger) = 5

(burger) + (2 beers) + (2 beers) = 9
5 + 2(2 beers) = 9
2(2 beers) = 4
(2 beers) = 2

The lower limit of integration is then

(burger) + (burger) – (bottle)
= 5 + 5 – 10
= 0

The numerator has the constant (bottle) = 10 and denominator is (2 beers)(hot dog) = 2(hot dog).

Hot dog is a placeholder variable so it can be replaced with x. This means the integral is:

The hard part is the integral of (sin x)/x from 0 to infinity. This is known as the Dirichlet integral, and there are many possible ways to evaluate it. I will present one method. First notice the following:

Taking the integral of both sides with respect to x from 0 to infinity gives the Dirichlet integral.

This double integral is not easy to evaluate. A standard trick is to change the order of integration. But this particular function over the specified region does not satisfy the conditions for Fubini’s Theorem. So one must carefully justify changing the order of operations. This is done in a paper by Paul Loya, “Dirichlet and Fresnel integrals via iterated integration” (JSTOR). Here’s the theorem:

And here’s the verification the integral satisfies the conditions of the theorem:

Loya, Paul. “Dirichlet and Fresnel integrals via iterated integration.” Mathematics Magazine 78.1 (2005): 63-67.

So now we proceed to change the order of integration.

We now need 5 times this value, which works out to 5π/2 = 2.5π. You might say that is “two and a half pies”! And that’s the answer!

References

Puzzle on Facebook
https://www.facebook.com/Mathematical.Mathematics.Memes/photos/a.114727355535937.1073741828.114667355541937/376605399348130/?type=3&theater

Math StackExchange on (sin x)/x integral
https://math.stackexchange.com/questions/5248/evaluating-the-integral-int-0-infty-frac-sin-x-x-mathrm-dx-frac-pi

Loya, Paul. “Dirichlet and Fresnel integrals via iterated integration.” Mathematics Magazine 78.1 (2005): 63-67.
https://www.jstor.org/stable/3219276
http://people.math.binghamton.edu/loya/papers/LoyaMathMag.pdf

Dirichlet integral
https://en.wikipedia.org/wiki/Dirichlet_integral

PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

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