What's in the suitcase?

We all love puzzles, right. I'm not talking about jigsaw puzzles, but puzzles of the riddling variety. These types of puzzles are like mental challenges that put our egos and abilities to the test, so that by solving them, we have gained some insight or disciplined skill which will help us wend our way down the path of our life. But certainly anyone who has ever been challenged with a puzzle knows the frustration that comes with each new puzzle. We sometimes find ourselves refusing the challenge or perhaps even running away before we get too involved even though we know the they are "good" for us, in much the same way that I fled from Carob as a kid when my mom thought it was a healthier substitute for chocolate.

It may start with the same letter, but it's NOT chocolate.

A good puzzle not only challenges us and forces us to try and consider things we otherwise wouldn't, but they're also beneficial because the spark our imagination and summon our curiosity. In the case of math puzzles (the BEST kind), they make math more palpable and interesting. They are miniature lessons in reason and independent thinking, which helps the victorious puzzle solver build self-confidence. The ability to solve these types of puzzles quickly and calmly can also save your life and the lives of hundreds around you, making you a bona fide hero!

If you've ever seen Die Hard with a Vengeance with Bruce Willis, you probably remember the famous "Water Puzzle" scene in which the characters played by Willis (McLane) and Samuel L. Jackson (Zeus) must measure out exactly four liters in less than five minutes to prevent a bomb from going off. No problem, right? Well, the catch was that they only had a five-liter jug and a three-liter jug. There is no convenient four-liter container anywhere to be found. The clock is ticking. Your life's at stake. Good luck . . . don't mess up!

Sorry guys, you're not allowed to "phone a friend."

This puzzle was not new to the movie, except maybe the exaggerated consequence of failing, but has actually has been around since the 11th century, when Bruce Willis was still an up-and-coming actor on "Moonlighting." Can you do it? Let's eliminate the bomb and timer aspect and see how we can go about solving this timeless riddle.

Method 1: Trial and error. This is perhaps the most primitive method that a monkey can be trained to do. It also requires real, physical buckets and liquid. Good luck finding a perfectly calibrated 5L and 3L bucket at your local Home Depot. But let's say you DO have the buckets and the liquid, if you keep filling and pouring, filling and pouring, you'll probably not ever produce the entire works of Shakespeare, but you will likely get your desired four liters. So what's "wrong" with this method? Well, it kind of takes as the fun out of the puzzle to begin with, and although it might get the job done if there ever was a real-life, stressful scenario in which your reptilian brain was all that worked, it requires a lot of time. Not the best method of the clock is ticking or the saber-tooth tiger is nearby.

Method 2: Google the solution. How resourceful! Cheater! You get NOTHING out of the puzzle except accelerating your carpal tunnel syndrome.

Method 3: Do the experiment in thought only and save yourself the cost of buying two buckets. Think the problem through prior to conducting your "gedankenversuch." Try it. You'll benefit from the process even if you're unsuccessful in solving it. As Oliver Wendel Holmes, Jr. twice said (they didn't quite hear him the first time), "A man's mind, when stretched by a new idea, never returns to its original size," which I guess is only a bad thing if you like wearing expensive hats.
Here's the answer: Fill the five-liter jug and use it to fill the three-liter jug. You now have two liters in the five-liter jug. Empty the three-liter jug, then pour the two liters from the five-liter jug into the three-liter jug. Refill the five-liter jug, and then top-off the three-liter jug (this will take one liter). Congratulations! You now have four liters in the five-liter jug.

Did you get it? Did you do it a different way? IS there a different way? Of course there is. Just like you can get from Austin to San Antonio, Texas by traveling south on IH-35 OR by traveling north (albeit it is a much longer, more scenic route), many things in mathematics can be solved in several different ways. The real mathematical challenge is finding the most simple, beautiful, elegant, and efficient solution.

An elegant and beautiful bathroom solution

Here's an alternative method. I'll let you be the judge of its pulchritude.

Fill up the three-liter jug, then pour it into the five-liter jug. Refill the three-liter jug, then use it to top-off the five-liter jug. This requires two liters. Empty the full five-liter jug, then pour the one liter that remains in the three-liter jug into the empty five-liter jug. Once again, refill the three-liter jug, then pour it into the five-liter jug. Congrats again! You've got yourself four-liters there partner.

Mathematicians never leave well enough alone. We're not content with specific solutions to contrived problems, but rather, we are interested in generalizing techniques and solutions to bigger and better problems. Sure, the process gets more onerous, and requires organized, discipline habits of mind, but they force you beyond simple "guess and check" methods and get you actually thinking! We're really not interested in jugs and water. We're interested in abstract patterns. Therein lies the true, abstract poetic beauty.

Some natural follow-up questions would be,

"Is it possible to measure out exactly ONE liter?"

"What if you had the same scenario, except with four- and nine-liter jugs?"

"Can you get all all quantities from one liter through 13 liters?"

"When is it possible, and when is it NOT possible, given to containers of capacity a and b liters to measure out exactly n liters?"

Answering the last question is what real mathematicians do. They solve things in the general case. They provide rigorous proof to theorems and conjectures, which is essentially decoding the natural universe and the handiwork of The divine creator!

Perhaps we should have listened to Samuel L. Jackson's Zeus character's advice and never have opened the suitcase in the first place. What a Pandora's box it turned out to be.

Give me n liters to go, and put it in my suitcase.