Predicting with confidence

If you've ever read time off of a clock, then you're familiar with something mathematicians call modular arithmetic. You might also think that modular arithmetic is science of counting all the possible ways you can configure an office using cubicles, and you'd be clever, but you'd be wrong. Modular arithmetic is simply ordinary arithmetic (adding, subtracting, multiplying, dividing, exponentiating) performed on numbers whereby you reduce the result to an equivalent number based on something called the modulus of the arithmetic. What does this have to do with a clock?

Well, let's say that it's 8:00 in the morning. You have to be at an important place in 5 hours, or 1:00 in the afternoon. Without much thought, you just did the following arithmetic operation:

8 + 5 =1

Writing it like that makes if feel strange and dirty and absolutely false, but yet it's true. . . in modular arithmetic. If you were in the military, or were one of those strange non-military people who prefer "military time," or a 24-hour clock, you would say that your important destination must be arrived at at precisely 1300 hours, or 13 o'clock. Because most people use the 12-hour clock, you're likely to get a few strange stares from people when you announced it, and you'd probably have to "do the math" for other interested, mathematically illiterate bystanders.

But it's exactly because clock values repeat them self in regular intervals of twelves that clock "arithmetic" is simply modular arithmetic with a modulus (abbreviated simply as "mod") 12. Because 8 + 5 really is 13, we can rewrite the above statement in modular form as

13=1 mod 12

We actually would read this as "13 is congruent to 1 modulus 12," which is a fancy way of saying 1300 hours is 1:00pm to most of us. Now that we've established this new math, let's say that we were big dorks, and one day we decided to not only abandon telling time on a 12-hour clock, but we wanted to keep track of the elapsed hours over a one year period. Imagine the strange looks we'd get if one afternoon 4 months later later, some stranger asked us for the time and we responded with "1841 o'clock . . . . on the DOT!" Would there be anyway to figure out the answer quickly without subtracting out full days at a time? Well yes.

What we'd really want to know in this case is the answer to the question 1841=? mod 24 which would give us the current military time, or, since we know it's in the afternoon, 1841=? mod 12, which would give us the current time p.m. The answer, as it turns out, will simply be the remainder obtained when dividing the given number by the given modulus, 1841/24 or 1841/12 here.

I turns out that 24 goes into 1841 76 times with 17 left over (1841 = 24*76 + 17). So 1841=17 mod 24. That means it's 1700 hours, or 5:00p.m. Using mod 12, we see that 12 goes into 1841 153 times with, you guessed it, 5 left over. So what time is it? Time to have that drink.

It turns out that MANY numbers are equivalent to the same number for any given modulus. For example, 5, 17, 29, 41, 53, 65, 77, 89, 101, etc. are ALL equivalent to 5 mod 12. For all such numbers that are equivalent to each other, we say they are elements of the same Residue Class.

Modular arithmetic works for other things too that occur in regular intervals. Take the world of competitive skate boarding. Imagine a shredder pulling off an AMAZING "7863" rotational trick. Since 1 rotation = 360 degrees and 7863 = 21*360 + 303, we know that so the skater did 21 full rotations plus an additional 303 degrees. Stated mathematically, we can write 7863 = 303 mod 360.

Modular arithmetic also pertains to the days of the week, with a modulus, of course, of seven. Perhaps you'd be interested in knowing what day of the week it will be exactly 1,000,000 days from today. Not that you'd be around to enjoy that day, seeing how its more than 2,739 years from now, but you're such a dork that you ponder these things. How could we do THAT calculation using modular arithmetic?

Let's start with today. Today is Friday (woooo hoooo! is it 5:00 yet?). From here, it requires a little more cleverness. Since 1 million is a power of ten, we will start the computations with the following, seemingly non-intuitve equation, and try to build up the left side, simplifying the result (mod 7) as we go.

10 = 3 mod 7

Feel free to verify this by dividing 10 by seven and finding the remainder. Now in any equation, even if it's modular, we are allowed to square both sides. Here we go.

100 = 9 mod 7

From here, we'll note that 9 = 2 mod 7, so we replace it in the above equation.

100 = 2 mod 7

We can now make a big jump to a million by cubing both sides. Since 100 cubed is 1,000,000 and 2 cubed is 8, we obtain the following.

1,000,000 = 8 mod 7

Now because 8 and 1 are in the same Residue Class, they are interchangeable. Performing the interchanging act we finally reach our desired equation.

1,000,000 = 1 mod 7

So how does that tell us what day of the week 1,000,000 days from today be? Well, the equation tells us that it will be the same day of the week that ONE day from today will be. Say it with me now, "If today is Friday, then tomorrow is . . . Saturday!"

Wasn't that a lot more fun than using a calculator? (Just say "yes.") Besides, our method works for numbers that are beyond the calculating capability of your calculator. If you don't believe me, tell me quickly what day of the week it will be 10 to the 100th power(that's called a "google" baby, a 1 followed by one hundred zeros) days from now.

If you you said "Wednesday, " you're calculator is lying to you.

Feel free to get back to me on that.

Answer For The Ages

Yesterday I posed a mathematical puzzle in which I challenged you to determine the ages for three children based on some loose, esoteric clues. Due to the overwhelming lack of response, I assuming that either no one read it or I was able to sufficiently stump you.

For those of you who DID give it a go, did you determine that the woman had twins? Did you figure out which one had blue eyes? Do you now know what the census taker ate for breakfast?

Let's explore now how the census taker, after a hearty morning meal of bacon, eggs, and grits, was able to complete his inventory of the house of the enigmatic mother of three.

For the first clue, the woman said that the product of ages of her children equaled 36. If we assume that none of her children are age "zero",as I used to muse about my 6-month-old daughter, and if we also assume that we're talking about integer ages of kiddos, then there are only a small, finite number of possible ages that fit the equation. A careful attempt can yield all eight possibilities. Here they are. Check my work. Make sure they're the same as yours.


Although it would be highly unlikely that the woman would have a pair of one-year-old twins and a another 36 year-old rug rat, but that still leaves seven respectable possibilities. It's obvious why the man had to ask for the second clue. But with the extra hint that the sums of the ages totaled the address of the house across the street, how does that help us? The census take actually got to SEE the number of that house, while we were kept in the dark. Well, it turns out that we don't even need to know the number of the house!! So why have that clue at all, right? To answer that, we'll need to look at what the sums of all the above combinations yield.


Do you notice anything peculiar about the sums? You should see that all but ONE of the sums are unique. The sum of "13" actually appears twice! This means that the house across the street MUST have been house number thirteen. Duhhhhh! Right? Why's that? If it's not obvious to you, I'll explain further. This fact has nothing to do with the numbers of the mathematics of the sums AT ALL. It comes from a clue from the puzzle scenario. Remember that the census taker needed a third clue in order to conclude his detective work. Had the sum been any of the other combinations, he would have had his answer, but because 13 appears twice, he needed further clarification as to which of the remaining two options were the actual ages.

Even though the woman was put out at the man's final request, comparing his annoying unrelentingness to her eldest child, and refused to offer her assistance any more, she unwittingly provided him with all the information he needed.

Do you see it yet?

Her children now had possible ages of 1, 6, and 6 or 2, 2, and 9. Because the woman has an eldest child, we can infer that she has a pair of two-year-old twins and one, eldest child age nine!

A pretty nifty piece of mathematical detective work, huh.

Well, stop the presses. Just like Andrew Wiles's origianl proof of Fermat's Last Theorem was later found to contain a flaw that was overlooked by everyone, so too does my little scenario contain a tiny glitch. My wife immediately pointed out that the ages could be a 1-year-old and a pair of 6-year-old twins, since even twins, apparently, are born ONE AT A TIME, so that even though they are the same "age," there is an OLDER twin and still an ELDEST child.

D'0h!

Andrew Wiles eventually fixed the error in his proof, and now it stands as rock solid. I'm hoping I can reconcile this tiny inconsistency as successfully as he was able to. Perhaps I can go back to the original scenario and somehow cleverly work in the necessary condition that all ages be rounded to the nearest year, or maybe I'll just rewrite it so that the status taker never even offends the woman at all, with her simply telling him, "two, two, and nine. Goodbye."

A Puzzle For The Ages

Here's a delightful puzzle to help you get over the Wednesday hump, or maybe it will get you stuck there. Either way, delight yourself in mathematical detective work to see if you can figure it out.

A member of a census organization is going door-to-door collecting information. He comes to a house where a woman answers the door. After introducing himself, he asks her how many adults live in the house. "Just me," she replies, as the screaming and yelling of children pierce through from the living room. "I'm guessing you have children," he astutely asserts, "how many?" "I have three, and they're about to drive me out of my mind. How many of them do you want?" She asks back. "Um, ma'am, if you'll simply allow ME to ask the questions," he rudely replies. "Now if you could please tell me the ages of your three children, I'll be on my way." Feeling slighted, the woman becomes irritated and less cooperative. "Absolutely NOT," she retorts. "You're gonna have to work at it now Mr. Census man. Here's a clue. If you multiply all their ages together, you get thirty-six." She then slams the door in her face.

The census man pulls out his calculator from his pocket and punches a few numbers, then knocks on the door again. When the woman answers the door this time, she is even more irate as the sounds of obstreperous children spill out of some hidden room in the house. "Sorry to bother you again, ma'am, but could you please offer me one more hint as to your children's ages? "If it'll get you off my doorstep, you should know that the sum of their ages equals the address number on the house across the street." BAM, slammed the door.

Determined to get this information, the census taker runs across the street and peers at the number on its mailbox. He again pulls out his calculator, punches a few keys, then scratches his head a bit. With a determined temerity, he knocks on the woman's door once again. "I'm so sorry to bother you again, but I need to have just one last clue and I'll be our of your hair for good." (Screams emanated from inside the house) "You're just like my oldest," she said "unrelenting!" Then she retreated back into the house and was gone.

Now the census man may not be suave, but he's no dummy. From her last, subtle clue regarding her child's temperament, he is able to discern the ages of each of her three children. Can YOU figure it out?

Answer and explanation tomorrow. But you can't read tomorrow's answer if you haven't mentally sweated over it today.

Returning policies

Right now at school, we're in the process of tying up all the loose ends to what has been a very productive and interesting academic year. As teachers try to simultaneously squeeze in one last test while preparing students for a final exam from which they know most students will end up being exempt, students are making summer plans. One of the most celebrated orders of business that students must tend to is turning in their textbooks.

A bitter-sweet thing, no doubt, this year our campus is returning to an old tried-and-true procedure for recollection: turning them into their teacher!!

The past few years, the gymnasium has been set up like a long corridor where students would bring all books to one specific class, then be released via a pre-determined schedule to the gym, where they would make their way through the "buffet" line, finding the appropriate pile for their individual textbook, and dump their cargo one book at a time. This procedure was very efficient, but it also had a few drawbacks.

First, the basketball team had to recreate in the other, smaller practice gym. Second, books inadvertently ended up in the wrong pile. For example, a student fresh out of failing his English class might put his spelling book in the Algebra I stack. Or, a lazy student might put both of the books he actually remembered to bring that day in the very first stack he saw. Third, there was no way to actually know if a student actually turned their book in. Because the books weren't scanned in as they were TURNED in, students who ended up owing $85 for a math textbook they probably never cracked could simply tell the authorities administrators that the "put it in the gym." It was creating a too convenient excuse for the apathetic, the dishonest, and the unfortunate victims of random textbook thievery.

This year, students must physically turn in their books to their respective teacher, and we teachers must document which students have and have not done this. Yes, it means more work for the teachers, but at least we don't have to go through every single page and add up a list of nickels that students owe us for every tiny stray mark or tattered pages like it used to be when I was a student, although we are still required to look for errant mustaches drawn on historical figures.

Being the collector of the precalculus and calculus textbooks for my students, this new-again procedure has afforded me the opportunity to interact individually with each student one last time before they hit the swimming pools. It also gives THEM a chance to finally write their name on the inside cover. I'm surprised how few of them ever took the time to do that originally. When I was in school, writing my name in my book was not only required, but it was like I was signing a contract for the course, making me feel a heightened sense of responsibility for its safe keeping and appropriate use. Back then, we also enjoyed looking at the long history of names that preceded ours. "Awwwww, I got a 'dumb' book," we might say if we saw that 7 years previously our neighbor down the street with the barking dogs and the appliances on his lawn had our book. "Alright!! Easy 'A' in this class," we would rejoice if we happened to get last year's Valedictorian's book.

It is interesting to see the various reactions from students as they hand over their heavy tomes of information. It's like a great burden, an immense weight has been lifted off their shoulders, and the stand a little taller afterward without their backpack weighing them down. It's also surpising how few of them even need to use the Kleenexes I offer them as they part with there "loved one." Most aren't even interested in saying their final goodbyes. I don't hear, "So long page 273, I'll miss you!" or "Adieu back-of-the-book-answers-to-odd-problems, you spent so many homework sessions with me." I don't even hear the failing student, "Adios stranger, I didn't even get a chance to know you."

What I DO hear alot, though, is "What on Earth is a book cover anyway?"

If they only knew . . .

You Go Girl!

Did you hear what happened over the weekend in the fascinating, thrilling, unpredictable world of thoroughbred horse racing?

At the 134th running of the Preakness Stakes at the Pimlico Race Course in Baltimore, Maryland, for the first time ever, track officials didn't allow the spectators to bring in their own beer, so the thrill and buzz seekers went to the NASCAR races instead.

A forever-lost Preakness Tradition

Also. . . .

A little girl beat the blinders right off of the 12 boys she was running against.

Coming out of the 13th slot, the 3-year-old Filly named Rachel Alexandra became the first female to win the event in 85 years, since Nellie Morse made the boys cry way back in 1924. Not even Nellie Morse's daughter, Nellie Flag, could accomplish what only 5 females have been able to do.

When asked after the race how she was able to leave all the men in her dust, Rachel, who could have truthfully gloated about how the rest of the competition were a bunch of losers, instead tactfully responded with just an enthusiastic whinny. The comment was later translated into, "All the men were horsing around, making dismissive jokes at my expense before the race, commenting on my manicured nails, trying to make me feel self-conscious about the white spot on my forehead, and my "unlucky" number 13, telling me that I belonged in the WNBA, which didn't even make horse sense. I just used all that energy to motivate me to prove to them, that although my name was not Babe Didrikson Zaharias or Ruffian, it WAS Rachel, and I had every right to be there." Indeed, it was a loooong whinny.

Finishing right behind Rachel was Mine That Bird, the Kentucky Derby upset winner, who made his characteristic late charge from the rear only to come up full length short of the galloping gal. When interviewed after his galiant but futile gallop, Mine That Bird was tight lipped in his response and gave only "genetic predisposition" for his noticably long face.

Now with two of the three major races out the way, each with a different winner, there will not be a Triple Crown winner again this year. Nonetheless, the Belmont Stakes, the final and most challenging race of the three, is shaping up to be as exciting as an NBA game seven between Magic Johnson's Lakers and Larry Bird's Celtics. After a gruelling test at the first two races, the showdown at the 1.5-mile track between the Kentucky-slighted Ms. Alexandra "the Great," and the Preakness-slighted Mr. That Bird will will be exciting to watch. Who will win the two-out-of-three battle of the sexes? Will Ms. Alexandra be the Billie Jean King of the Belmont or will Mr. That Bird be the Peter Sellers of the classic British comedy, or will it be more like the upcoming collaborative album by Ludacris and Shawnna?

Although neither horse will have a chance to win the coveted "Triple Crown Oats Bucket" this year, there is still possibility that a Triple Crown winner will be crowed. How can that be? It turns out that Mine That Bird's little jockey in the Kentucky Derby, Calvin Borel, didn't ride him again in the Preakness. Instead, Mr. Borel mounted a different horse (with the help of a ladder) for the second race. That horse was Rachel Alexandra. This begs the question: how did he KNOW to do that? Is Mr. Borel so talented that he can race ANY horse to victory, or did he just get lucky in sitting on two animals who would have with a 120-lb sack of flour on their backs? I guess we'll just have to get the answer to that question in a few weeks at Belmont. It would be interesting, although highly unlikely if a 120-lb sack of flour existed and also entered the race, and if Mr. Borel mounted a thoroughbred by the name of "Born To Lose," or "Congenital Defect."

I wouldn't bet on it.

Productive Day

Some simple things I learned today:

• John Michael Montgomery's brother IS in the Montgomery-Gentry duo, but he's actually the OTHER guy from who I thought he was.
• Even make-up games can get rained out.
  
• It doesn't matter how much compressed air you use after a haircut, you'll never blow all the tiny pieces of hair off your body.
  
• Taking a shower with your watch on feels as awkward as sleeping with your socks on.
• You NEVER want to run out of chlorine tablets for your septic gray-water.
• It's okay to take a nap in the middle of the day in the middle of a rhythmic downpour, as long as you're not wearing your socks.
• Any attributes that qualify you for delivering a graduation keynote address are the same attributes that qualify you for an honorary doctorate degree.
• It's no fun to follow basketball if your team's out of the playoffs, nor is it fun to watch baseball when your team is the "Bad News" Astros.
• Some coffee houses DO still understand an order as simple as "Tall Black Coffee."
• Macaroni and White Cheddar is nowhere near as delicious as regular mac & cheese.
• Dead bees can still sting.
• The quality of my golf game is inversely proportional to the amount to time I practice it.
• The same is true for everyone who is at a driving range on Saturday morning rather than on the links itself.
• "Barack" moved up 10,126 places last year on the list of "popular boy names." Unfortunately, it is still in 2,409th place.
• Making homemade toothpicks is a waste of time.
• A $500 diamond ring can bring a smile to woman's face.
• Plastic lag screws are a fast, cheap substitute to using Tapcon Masonry screws. By the time the box falls of the exterior of the house, the cable man is long gone.
• It's best not to try to guess the mysterious liquid accumulated at the bottom of the kitchen trash bin or how it penetrated the "Force Flex" tough Hefty bag.
• Did I already mention that you NEVER want to let your septic run out of chlorine tabs?
• Fruit bowls are just a place to hold rotten bananas.
• Anything you want in life, you can purchase a "Hannah Montana" version of it.
• Even though the traffic signs say, "Watch for Water on Road," I notice a LOT of don't even try.
• Sonic Drive-In boasts 168,000 possible drink combinations, one of which is "Large Tea with sweetener in a cup with ice and a lid and a straw."
• Lifetime warranties on sunglasses don't cover you losing them, 'cause, technically, they're still "alive" somewhere, just not on your face.
• Taking naps makes you sleepy.
• Disney character shirts make my look Goofy.
• Some landscaped commercial areas have crosswalks that lead directly up to them for no apparent reason.
• Few things are as ephemeral as a clean childrens' playroom.
• "Pink Panther I" and "Pink Panther II" are some of Steve Martin's best work.
• Blogging about underwear is less interesting to readers than blogging about my drab, boring "Mathematical Musings."

Look under THERE!!!

Did I just make you say "underwear?!!" Tee-Hee.

Here's a short poem to commemorate.


I met a man one day
who had a notion in his mind
to give me all his underwear
at least, all that he could find.

I thanked him very much
and said, "That's very kind and all,
but I have plenty in my drawer.
Besides, you're kind of tall."

"Your underwear won't fit me,
only cover up my FACE :0
That's not the way they're 'possed to fit
That's not their common place."

"Besides, yours have tears in them
and holes inside and out .
You should, instead of offering them to me
just go ahead and throw them out."

"Okay," he said
as he put them in the trash.
"You wouldn't want 'em anyway,
'cause I've got this ugly rash."

So off he went away from me
to carry on his life,
to the one who washes all his drawers,
his kind and underwear-buying wife.

Mathematical Musings: XVII

Again, another installment of the silly things I've said in class, although I'd deny every bit of it.

• Hurry up and wait your turn.
• I scheduled my alarm for 2:17 so that I’d remember when to turn my alarm off.
• Student: Look Mr. Korpi, I have an idea. Since the schedule this week is all messed up, why don’t we split the assignment up in two. We can do part today and part tomorrow. I think this would be a good compromise. Korpi: What are you—the Henry Clay of Calculus Class
• Today is the first day after the second day before the next day that comes after the previous day last time.
• Please live each day to its fullest. Another Wednesday won’t come around for another week
• When it comes to teaching Calculus, I’m like Pavlov’s dog. When I hear the bell to start class, I start salivating.
• My wife and my favorite time of day is dusk: when our son goes down. He is such a dynamo.
• SURE we can watch a movie next time in class. However, I don’t have a VCR, so we will just have to sit around and stare and the cassette tape. I hope it’s a suspenseful movie!
• I love to get up every morning and look at the photos in the local newspaper. After all, it is the journalistic equivalent of a children’s picture book.
• f I was any stronger in will, I might have the power to consider saying “No”—to nuclear arms.
• Today after school, I will be giving an AP review. If no one shows up, I will still give the review, because that’s just the way I am—a man of my word. Please feel free to walk by and laugh at me through the window.
• The next time I have surgery, I’m going to pay the extra cost to have a STERILE room. It’s money well-spent. And that extra $5 to be off the floor, the cost-benefit-analysis has been DONE, and you should DO it! Unless you like hospital food.
• Student: when is all this weather supposed be good. Korpi: Tomorrow . . . . . . . If you think rain, thunder, mud, wind, and cold are good.
• Welcome 4th period . . . . . . I’ve been expecting you.
• Hurry, hurry on in to class. Seats are going fast!
• Ouch!!!! That look like it hurt worse than dividing by zero!
• This world needs more saber-toothed tigers.
• It appears that all you need to do to be immortalized on the front of Beer-lover’s T-shirts is to come up with a divine sanctioned quoted condoning the consumption of the frothy ale. The only competitor is Ben Franklin. Here’s my contribution. "On the sixth day, God also made beer; this is the primary reason he did nothing on Sunday."
• It’s a sign of our times, and perhaps our self-incrimination, that our 4 \-year-old son knows the meaning of being “voted off” of a television reality show.
• Student: How long will the test take today, Mr. Korpi? Korpi: Well, at least the whole period . . . . or less.
• OK, I know we’ve worked a lot of examples today, so this will be the last one. . . before the next one.
• Feel free to burn me in effigy over the weekend.
• Today as you work on your semester review, think of me as Mr. Radio Shack: you have questions; I have answers. I'll be over here behind my computer chatting with Trekkies.
• Because I accidentally made the final exam a little longer than usual, you will unfortunately not be able to finish it. But the good news is, I will still grade the ones you don't do.
• Man, I feel like Santa Claus in a shopping mall around semester review. This long line of students at my desk waiting to talk to me.
  

Blloggoggoggaggoggagghogg

Do you see anything wrong with this sign?

Could it be that the depicted Indian is actually a Cayuga and NOT a member of the Nipmuc tribe? Perhaps the NIPMUC INDIANS ESCHEWED THE USE OF ALL CAPS? Maybe the sign incorrectly overstates the power of the Massachusetts Turnpike Authority? Maybe the Massachusetts Chamber of Commerce had nothing to do with the sign.

NO, to all of the above. There IS and error, but it's much more subtle.

Did you catch the misspelling? (Yes, "misspelling" has two "s"s).

Believe it or not, Lake "Chargoggagoggmanchaoggagoggchaubunaguhgamaugg" should actually be spelled Lake "Chargoggagoggmanchauggagoggchaubunagungamaugg." That means there's actually TWO mistakes, not just one.

Now before you demand that the typesetter at the sign company be fired for being so blatantly incompetent, realize that he CORRECTLY place 43 of 45 letters correctly, which means that at almost 95%, he's likely to have been the salutatorian of his typesetter class. Think you can do better?

Besides, what kind of name is Chargoggagoggaggoggmangg . . . whatever, I'm not getting paid to typeset (it that's even how they MAKE signs, nor am I paid to blog)? As you probably guessed, it is from the Nipmuc (not Chayuga) language for "unnecessarily big name for a relatively small body of water," at least that's what I deciphered it as after my extensive research quick Google search of Nipmuc language. But after a much smarter search of Google, utilizing the "always accurate" Wikepedia.com, I realized that I was off in my translation (so much for using "context clues" like my English teach had taught me so many years ago). Apparently, the translation is different, but no less humorous, than my primitive interpretation. What some "expert" linguists (and conspiracy theorists) believe the name means is "you fish on your side, I fish on my side, and nobody fish in the middle."

What selfish indian fisherman! I why exclude indians with the advanced skill of "boat making" from participating in the "feeding of their belly?" Imagine if I took the same approach to my AP Calculus class, renaming it "AP callcallullaaccittucctticcuus" which would translate in my own, made-up language called "Grumpyoldmanese" into the phrase "Smart people take, dumb lazy people don't take, and everyone else give my $20." Those Nipmuc indians were the original semantical warriors.

Most politically correct experts of the Nipmuc language, whose name is Frredd, claim the name, which is incidentally the longest name of any official place in the Americas, not counting the unofficial name that many students give to calculus of "Calc@##$%&*!!@#*$%$%^&#@*&*%$$#@^&^@#*&^%##**&^@#$%!**$%&*@#$%! . . . (#*&$$^@&*%(!@#*$&%&$&lus," seem to opt for the less curmudgeonly-sounding "Fishing Place at the Boundaries -- Neutral Meeting Grounds," which makes the lake sound more like a meeting for a boring convention rather than a recreational fishing hole.

Then there are some boring, uniteresting people in the nearby town of Webster, Massachusetts who simply refer to the lake as "Lake Webster." These are the same people who eat Hamburger Helper for dinner every single night and know not when (or at what temperature) to serve white wine, primarily because Boone's doesn't make white wine. That type of oversimplifaction not only denies the lake of its cultural legacy and therefore its place on the map and blogs across this great land, but it is the same attitude that failing students of mine have when the approach Calculus as if it were entitled "Adding single digit integers on you calculator"--doomed to scornful looks, derision, and a good laugh. Good thing that THESE type of individual are not the official people in charge of things.

The good news for the lake, though, is that Massachusetts OFFICIAL officials have admitted to the two misspellings and have vowed to have the signs changed to reflect the correct spelling of the lake. State officials have already begun interviewing for the job of making the new sign, beginning with the Valedictorian of the "typesetting class," a Thai student with the successful completion of a sign bearing the the native name for Bangkok, Thailand on his resume:

Krungthepmahanakonbowornratanakosinmahintarayudyayamahadilopo-
noparatanarajthaniburiromudomrajniwesmahasatarnamornpimarnavata-
rsatitsakattiyavisanukamphrasit

Gone to the Dogs

Imagine that your dog Rover wants to go see friends in Seattle. What do you do? Do you let him venture off like a stray, navigating his own way through the concrete jungles, relying on his internal compass and instinctual methods for finding food, shelter, etc.? Do you arrange for the dog friend in Seattle to meet Rover halfway? Do you take him yourself, then sit by awkwardly as the third wheel in this dog reunion? Do you just put him on a plane by himself and say, "Arf voir."

Wouldn't it be great if, assuming Rover can save up the airfare, you COULD just put him on a plane to his desired destination? Perhaps you can do that now. But what if Rover didn't have to fly in cargo, but rather was treated like a regular fair-paying passenger, getting those tiny pillows and pack of peanuts?

I can see it now. You take Rover to the terminal, then had over his leash to a "Pet Porter," who introduces him to Fido, Fifi, and Ferdinand. As you leave your beloved pet, you have the comfort of knowing that Rover is in good hands, and that his safety is all but guaranteed as all dogs are subjected to metal detectors and stool samples. As Rover and his new friends await to board their own private 19-passenger Beechcraft, they are walked, watered, and combed.

As it comes time to board, they are individually walked down the red-carpet gate where they enter the plane, greeted with a friendly "Bark, Bark!" and a quick sniff from their Captain and his St. Bernard co-pilot. They are taken to their window seat, where they enter their own private pet carrier, that is safely fastened to the floor. As the plane departs, a "Pet Attendant" walks up and down the isles, assuring the safety and comfort of all the animals, making sure none make any noise while the "No Bark" sign remains lit. All through the flight, the animals relax in comfort in a pressurized, climate controlled environment with fresh air constantly circulating. "Paw"sengers, as they would be called, have their choice of the in-flight movie: "Beethoven," "Beethoven's 2nd," "Beethoven's 3rd," "Beethoven's 4th," "Beethoven's 5th," or "Beethoven's Big Break." In the event that Rover needs to use the lavatory, an attendant escorts him to the back of the plane, where yesterday's newspaper covers the floor. Thank goodness for that circulating fresh air!

Upon landing, Rover and the other pawsengers are given the local time, weather conditions, and locations of historic fire hydrants. Once they remove their carry-on bag containing their squeeky toys and Milk Bones from the overhead compartments, they are individually escorted to the terminal, where they are greeted by their pet friend and their pet freind's family. For dogs without a host family, rental cars are available for them to chase to their desired location.

Wouldn't it be great if such a service for pets existed? It would be nice, but probably not realistic, since, well you know, a flight service just for dogs and cats? It's doubtful if there is a large enough market for such an exclusive service. That's just a ficticious scenario right out of "Beethoven's 10th," right?

Actually, it describes, for the most part, the real service soon to be provided by a new company called "Pet Airways." For $149 for a one-way ticket, owners can book individual flights for either their dog or cat. Currently there is no option for first-class, but cats and dogs shall be separated within the aircraft, for obvious unknown reasons.

Now before you think that this service is "for the birds," be forewarned that the airline currently does NOT accomodate our feather friends, standing behind their current policy of "Birds Can't Fly, silly!" (Note, that is not actually the company's slogan). As of right now, birds are only allowed to follow outside the plane. The company, though, is NOT against expanding their "pawsenger" line to animals that don't have paws, they just have to find a cute and clever name to call them first.

Let's help them out. What type of animals would YOU allow on your plane, and what would you call them. I'll get you started:

Allow Gerbils on board. Call them "passengerbils."

I'd still be a Pawsenger, idiot!

How to keep 'em hooked.

As the school year draws to a close, students and teachers alike are excited, to say the very least, about being free from each others' bondage. With only three weeks left of school, some have resorted to crossing days off there calendars. Others are crossing off minutes. Yes, I DO have such a calendar.

With the disruption the Swine Flu break brought us, these last three weeks are even more stressful and eventful than usual as we try to finish AP exams, squeeze in those darned TAKS tests, not to mentioned Prom, awards assemblies, and end-of-the-year banquets for every possible student group on campus (excluding the small start-up organization called "Students Against End-Of-Year-Banquets").

As a teacher who is too lazy to plan nothing, it's been increasingly difficult lately to impart my mathematical knowledge, and as usual, I am relegated to the role of "coaxer," "cajoler," desperately trying to pull each and every student across the finish line. Threats of "holding them in low esteem," or "taking away their birthday," are of little consequence. Even for some of the most conscientious senior students, the threat of a string of zeroes in the grade book becomes a moot motivator, as senior class rank came to a close at the end of the last grading period.

For those students who are still awake in class, it has become amusing to hear their private conversations that are taking place while I'm lecturing about volumes of solids of revolutions and central angles of pie charts. They're giddy over prom, excited about their new tan, and sad that the television season is showing its finales. Some are still giddy over the math itself, or maybe it's just me being extraordinarily silly, entertaining, bombastic, sarcastic, and "creepy." With ears in the back of my head (and behind my kneecaps. . . . . nevermind), I can still hear all the individual conversations over the roar of my loud, exaggerated voice and dramatic antics. Occasionally, I hear something so salacious or something I can deliberately misinterpret that I stop myself mid-leap, mid-sentence, mid-equation and draw attention to the conversation in a way that leaves the conservationists feeling embarrassed, guilty, and impressed with my ability to do things like that. Everyone else just temporarily stops their conversations and gives each other that "look" that says, "Wheeew. I'm glad he didn't mortify US," and "What a crazy, psycho with strange-looking ears behind his kneecaps we have as a teacher."

But I digest, especially after lunch.

I think the real key for any teacher this time of year is to realize that although you WISH the students would respect you and owe you their attention and best efforts by now, that it is still our responsibility to capture their attention first, to instruct second. Additionally, a savvy teacher does not give students the slightest chance to "drive the bus," but must rather set the tone from the moment the students enter the class. Keep them on their toes. Keep them off balance. Keep that fire stoked beneath their little tootsies.

But there's a method to my madness.

If that means having them think that you forgot to take your medicine that morning, so be it. How fun is it to be totally understood by everyone anyway? If it means not actually taking your medicine that morning, be careful. How fun is it to be misunderstood AND depressed with a bad heart and a bad case of foot fungus?

No. We definitely want to life to teach another day. Another year. It's why we got into this business to begin with. But that doesn't mean we don't enjoy our summers!

But I digest . . .

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.