I don’t consider myself a movie aficionado, but I decided to rent *Die Hard 3* under the motivation of the talented cast. Bruce Willis and Samuel L. Jackson – for $4, I’m in.

Later, after sitting through many scenes with explosions and fire and loud noises galore, I watched our protagonists wander into a park, where they are offered yet another life-or-death proposition: Measure out four gallons of liquid.

Yes. I, too, thought that this movie climax was a little lame, but then came the kicker: Bruce and Sam had to measure four gallons… using only one three-gallon jug and a second five-gallon jug! Of course, our boys, being the undercover math heroes that they secretly are, end up weathering the high pressure and saving the day.

So how did they do it? How do five and three somehow make four? I’m a little slow, and I was in a slight food coma from munching on nachos, so it took me a little while to sort it out, but the solution is rather simple once you get to it. For those who would like to ponder this little brainteaser for themselves, please skip down to Mr. Jackson’s *Star Wars* portrait.

For the rest, the basic idea is as follows:

1. Sam fills the five-gallon jug with liquid and uses that to fill the three-gallon jug to the brim. Bruce is now holding these three gallons.

2. Bruce pours out the three gallons—but Sam’s five-gallon jug still has exactly two gallons remaining! Sam pours these two gallons into Bruce’s jug. Note that Bruce’s three-gallon jug now has exactly one gallon of empty space.

3. In a final heroic step, Sam fills up his five gallons and pours water into the 2/3-full three-gallon jug until it fills, thereby eliminating exactly one gallon from his five-gallon jug.

Five minus one is four, and there you have it, folks: four gallons. According to the Mayo Clinic, that’s about six days’ worth of recommended water intake, so Bruce and Sam could call up four buddies and survive a day in the park without fear of any dehydration. The sad part is that even this would probably make for a better plotline than that of the actual movie. Adding a lightsaber wouldn’t hurt, either.

My *Die Hard* days ended there, but at least I managed to milk a homework assignment out of the experience. Enjoy!

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I wonder if the spam-eating machine ate my comment on this post, since it seems to have disappeared. Mostly I was expressing disbelief that hollywood actually did that. Thanks for the amusing post!

So, as mathematicians, we can’t help asking ourselves: what is the general solution? what sets of containers of what sizes will allow the measurement of exactly n gallons in general?

It’s an interesting problem and here are some answers to these general questions, as well as a java applet for playing with the problem, and a crazy triangular lattice interpretation!

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Prof. Kateon October 5, 2008at 1:41 am