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.

A man you should know about

There once was a man name Leonhard Euler (pronounced "Oil-er"). No other man in the history of mathematics, with perhaps the only exception being Newton, has his named attached to so many ideas, postulates, proofs, and former NFL football teams than the Master Euler. Without a doubt, he was the most prolific writer of mathematics ever, his total works filling at least 100 volumes (and still counting). He averaged about 800 pages of new work each year during his long, productive life. It is said that it would take an entire volume itself just to publish the table of contents for his works.

He left his mark on virtual every branch of mathematics in such diverse fields as number theory, analysis, hydrodynamics and mechanics, topology, cartography, astronomy, and even dabbled in the seemingly unrelated fields of science, public affairs, philosophy, and even theology. Without his passion for discovery, the world would be much less advanced than it is today, and common mathematical symbols like i (the imaginary unit, equal to the square root of negative one), π (pi ~3.14159...), f(x), and e (called Euler's number, or the natural base ~2.71728...) would likely look more like numbers rather than those sneaky symbols that look like letters. Because of Mr. Euler, I can write my name like this:

Euler's most important contributions were so vast and numerous that if I referred to something as simple as "Euler's formula" or "Euler's theorem" people would either look at me like I was a dorky math nerd, or if they were a dorky math nerd themselves, would chuckle before incredulously saying, "which one?" (We mathematicians love the humor brought about by an elliptic statement, like the previous one, or hyperbolic statements, like, "Have you ever heard of Korpi's theorem??") Depending on the context in which you refer to the greatest mathematician of the 18th century, his "formula" or "theorem" can take on many, many different meanings. Just in mechanical physics alone, one has Euler angles--which specify the orientation of a rigid body, Euler's theorem--simply stating that every rotation has an axis, Euler's equations for motion of fluids, and the Euler-Lagrange equation--arising from the calculus involved.

Despite what you learned growing up that there are only two types of numbers in the whole-wide-world: the number Zero, and numbers that aren't zero, there are actually many types of numbers. In fact, when it comes to Euler, there are both Euler numbers and Eulerian numbers, and Euler's number (the number e already mentioned), and they are NOT the same thing. Some numbers resemble the three previous types, but are not those types. These numbers I have personally named Euleresque numbers. Euler's study of the Bridges of Königsberg can be seen as the beginning of combinatorial topology (which is why the Euler characteristic bears his name, but to make matters worse, the Euler characteristic is sometimes called the Euler number too!! I told you it depends on the context). The Bridge problem itself lends itself to defining what's called an Euler walk.

Can you cross all yellow bridges only once??

If you were to ask any advanced calculus student what the "Euler Formula" is, they might give you two different answers. On one hand, it can refer to the equation that defines the exponentials of imaginary numbers in terms of trigonometric functions. But there is another "Euler's formula" that (to use the modern terminology adopted long after Euler's death) gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers. Wow! That's a mouthful. My BC calculus students would tell you that Euler's Formula is something entirely different.

They would tell you that it's the equation that proves the existence of God! That's right, they have a good teacher. Sometimes called Euler's Identity or Euler's Equation, it can be derived from the first context of Euler's formula above evaluated at π, we can arrive at the following equation via infinite series:


Why does it proves God's existence? Well, it would fall under the Teleological argument of creation by design. The equation is too perfect and beautiful, not to mention TRUE, that it couldn't have existed unless it was created by a god, by THE God. The equation itself contains the three basic arithmetic operations occuring exactly once each: addition, multiplication, and exponentiation. The equation also links the five fundamental mathematical constants: 0, 1, π, i, and e itself. Like the likelihood of a finely-crafted, precision Swiss watch just coming together by a random arrangement of its parts, or the probability of the monkeys at typewriters producing the complete works of Shakespeare, the equation itself had to have had a creator, a divine one.The creator is God. Euler merely discovered it.

Are you still not impressed with this guy?

It was said of him that he "calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind." But don't think that an industrious brainiac was all seriousness. He also had quite a sense of humor too. For instance, late in life he went blind in one eye, and when asked how it would affect his math studies, he quipped, "Now I shall be less distracted." Talk about making lemonade out of lemons. In fact, he soon thereafter became totally blind in both eyes for the last 17, and MOST PRODUCTIVE, years of his life. This gives entirely new meaning to being "in the dark" when it pertains to mathematics.

Euler's powers of memory and concentration were incredible and quite legendary. He could recite the entire Aeneid, all 12 books!! word-for-word. I have trouble just remembering how to spell Aeneaid. A guy who spent so much of his waking hours in his study discovering and writing mathematics still had time to father 13 children with two different wives. He was not troubled, however, by all the distractions, interruptions, and little league baseball games. In fact, he did most of his work with his children playing at his feet, never kicking them away. His mind was a human calculator, capable of doing prodigious calculations in his head, which he increasingly relied on after he went blind. Apocryphal evidence tells the story of two of Euler's students who had independently summed seventeen terms of a complicated infinite series, only to disagree in the fiftieth decimal place; Euler settled the dispute by recomputing the sum in his head.

Genius like Euler comes once a century, if that, and although he and I are so very different, we do have one thing in common besides our sense of humor . . . we're both Yankee fans.

Come on, Ump. A blind ma . . . I could have made that call.

Life-saving Math Skills

Suppose that . . .

You're stranded on desolate island in the middle of the Pacific Ocean with nothing but an unlabeled canned good and your TI-83 graphing calculator housing dead batteries. A rescue boat filled with Humanitarian Mathematicians finds you and offers to rescue you if you can answer the following question:

Which is bigger, the square root of two or the cube root of three?

What an irrational question involving two "random" irrational numbers! Why on earth would answering a question like that be required for rescue? Remember, these are not just selfless Humanitarians, they are Mathematicians trying to save the world one stranded, computationally-illiterate at a time. Their boat. Their rules.

Do you want to rely on the 50-50 odds of guessing *gulp* incorrectly? Are you brazen enough to ask the Mathematicians if they have four healthy AAA batteries you could borrow temporarily? Is there a way to definitively determine the answer just by doing a couple of quick calculations in the sand? Your life depends on it.

Your instinct might tell you to go with "cube root of three" since, as Shel Silverstein wrote about in "Smart," "three is more than two!" But with any good question posed by a boat load of mathematicians, the answer may be counter-intuitive. In that case, perhaps going with "square root of two. . . final answer?"

The problem is that, although three is bigger than two, taking the cube root of it takes more of the number "away" than taking the square root of it. Therefore, although the two is a smaller number, there is more of it "left" after the indicated operation. Since the numbers two and three are relatively close to each other (with only an infinite number of real numbers between them), the answer is not as obvious as one would think.

Luckily, there is an easy way to answer the question definitively correctly, but it does require a little cleverness, insight, and talent, the same things that enabled Picasso and van Gogh to paint their masterpieces and the same qualities that allowed Bach to compose and Keats to write. The solution is beautifully efficient, artistic, and like a good magic trick, is easy once you know how.

So clear of a space in the sand and grab a shard of seashell and let's work on getting you off the island and into that boat with those creepy Humanitarian Mathematicians.

1. First Let a equal the square root of two and let b equal the cube root of three. Working with single variables will be much easier than working with radical expression .
   
2. Next, rewrite each radical expression as an equivalent expression involving a number raised to a rational exponent.
3. Now comes the clever part. We will change the value of each expression by raising each number to the power of n, where n is the least common multiple of the bases, two and three. In this case, n is 6.
4. Simplifying each new expression using basic rules of exponents (the boatload of mathematicians are very happy that you know when to add exponents and when to multiply them, for sure), we arrive at two easily comparable integers.
5. Now for sure, it is obvious that 9 is greater than 8, so b to the sixth is greater than a to the sixth.
6. Since the numbers are positive, we can now take the sixth root of each side (raising to the 1/6 power) without altering the inequality.
7. Now we arrive at the solution we originally desired. The answer is now so obvious. Since b is greater than a, the cube root of three is in fact bigger than the square root of two.

As you stare at your beautiful artwork, your masterpiece in the sand, you are filled with the satisfaction that you're a genius, a very hungry, dirty, unkempt genius, but a genius nonetheless. As the Humanitarian Mathematicians congratulate you and welcome you on to their boat, you hesitantly leave your creation behind, undisturbed in the sand.

You are saved, hurray. Surviving the boat ride to civilization with a boatload of mathematicians is another story altogether, but at least you saved yourself through your own ingenuity and persistence. It could have been much worse, you know. They could have asked you what was inside the unlabeled can!


By the way, in case your batteries are dead in your calculator too, the decimal values of the two numbers are surprisingly close.

A day to remember

Today is a very special day for anyone looking for an excuse to drink margaritas, eat chips and salsa, and give thanks for the Swine Flu "epidemic." Today is Cinco de Mayo, which to those unilingualists out there means, "The day after the fourth of May." With more independence day celebrations than Zsa Zsa Gabor, today Mexico has reason to raise their blue-cloth masks high into the contanimated air and celebrate their pride and heritage. Although today certainly is no "diez y seis de Septiembre" (translated as "a week and three days prior to September 26th"), which is the Mexican National holiday commemorating their official independence from Spain, it does signify the huge victory of the small Mexican town of Puebla's victor against a much larger, better equipped, well-dressed legion of French soldiers smoking fancy cigarrettes. It's essentially their "Alamo," except they actually won (only to lose later, whereas we lost, only to win later.)

The day is still of significance in parts of Mexico, including Mexico City, which, ironically, is where the French eventually won the war and placed French Emporer Maximilian I on the Mexican throne. Rather than celebrate their defeat, enthusiasts celebrate their neighbor's victory by dressing up in hot polyester uniforms and reinacting the entire Puebla battle under the smoggy air around them which simulates rifle smoke.

Just up the road in Seguin, Texas, the town named after the Spanish General responsible for inspiring his Mexican troops that day back in 1862 to "wake up and fight (translated)" the kids who are home due to the Swine Flu are likely celebrating the day out in their yards flogging pinatas made to look like pigs and French Legionnaires.

In St. Paul Minnesota (see left), thousands of miles from Puebla, Mexico, little ninos and ninas are marching through the streets, ya know, wearing sombreritos (little sombreros) with what appears to be their hands handcuffed behind their backs doing folclorico dances.

Even farther north, across the other border (where they speak French) in Vancouver, they are celebrating Cinco de Mayo (and the French defeat?) via a skydiving exhibition. Apparently, the skydivers dress up as their favorite candy treat and jump out of a plane decorated as a pinata. Upon landing safely on the ground, the onlookers storm them and shred their costumes to pieces! For real! I'm not making this up.

In Portland, Oregon, more than 300,000 people attend a festival dedicated to this day. During the festival, people of Mexican heritage sell their crafts and wares while continuous Mariachi music plays loudly in the background while attendees stumble around drunk on Mexican beer trying to sing along to the Mariachi music with slurred lyrics in a language that is NOT Spanish. A similar spectator event is held at Civic Park in Denver, Colorado that also includes a green-chili cook-off and a holding facility play area for kids.

Even the British get in on the celebration in the Cayman Islands. At the local Hard Rock Cafe, you can drink enough discounted Corona beer until you get the courage to enter in their "Cinco de Mayo Air Guitar competition extravaganza!" Aside from the discounted imported Mexican beer and the day on the calendar, the event has nothing to do with Mexico, although you DO get
extra points in the competition if you throw in Air trumpet while wearing a giant black mustache and hoot a couple of "Aye, aye, ayes"

Last year's winner celebrating Hendrix style

As for me, I think I'll celebrate the day myself in a more subdued, yet honorable and flavorful way: Mexican food and a Negra Modelo(s) for lunch, followed by a looooong siesta. Thank you General Seguin and Swine Flu for making it all possible.

May everyone's tacos and enchildas of today not be the hearburn and gastroenteritis of tomorrow.