## Author Archive

### It’s a Threeven Day!

March 3, 2011

Happy 3/3 everyone!

I just graded a bunch of proofs that √3 is irrational.  The proofs had a lot of holes in them.  This didn’t surprise me too much, in large part because the students weren’t math majors; rather, it was for a liberal arts math class taken largely as a gen ed requirement, and the whole proof by contradiction thing is really pretty scary and abstract for most people the first time around under the best of circumstances.

But actually, even when I’ve assigned this to math majors, they struggle.  They can have the proof that √2 is irrational right in front of them, be instructed that instead of even numbers they want to look at multiples of 3, and despite my Find and Replace instructions, they still don’t understand what to do.  The most common mistake is to replace “even” with “odd”.

In some ways this doesn’t surprise me, but in some ways it does.  Why is it such a conceptual leap to go from 2 to 3?  It’s a HUGE leap for many people.  And so I was pondering this while grading, and Batman suggested it might be because we have a special word for “divisible by 2″ but don’t for “divisible by 3″.  So you get, what, 10 years of reinforcement that there is just this one special way to divide the integers, and it doesn’t generalize.

What we need is a new word for these numbers.

And fortunately we have one:  threeven.  So 0, ±3, ±6, ±9, …  are all threeven, and the rest are…umm, not.  (Maybe we need two new words).  This word isn’t mine or even Batman’s; it actually was suggested by one of his students in response to this exact same problem.

As a bonus, it generalizes:  there’s fourven, fiven, sixen, seven-en (sev-en? )…as far as you want.     Which, admittedly, might not be very far but it still makes for a smoother sounding proof.

Happy threeven day!

### Winter Math Jokes

February 2, 2011

It was a crazy January (with an inadvertently extended sabbatical, thanks to the ice storm down south at the time of the Joint Math Meetings!) and now February is coming in like, well, February.  Rochester is in the middle of a winter storm, and though it doesn’t quite seem to be the WINTER STORM that the forecasters predicted, there’s still a respectable amount of snow and ice.  Leading to conversations like these:

Last night, looking at the closings online:

Person 1 : Wow, they’ve even closed all the Curves gyms in the area, except for one that’s on a delay.  They list them all separately — that’s weird.

Person  2: Isn’t that a complicated what of describing it?  If they just made one announcement it would be Simple Closed Curves.

*************************

And then this morning…

Person 1: Can you take the kids to school tomorrow?  I’m giving an exam and want to allow plenty of time to drive slowly if the roads are still icy.

Person 2: Is that a Margin of Terror?

### Mandelbrot Video

January 3, 2011

I was thinking of all the things I meant to post in 2010, that I diligently saved, but that became less timely as time went on.  D’oh!  Crucial mistake, since it turned out the alternative was…blankness.

So I thought it might be fun, at least in a New Year Cleaning sort of way, to post them.  And I thought I should call it Ten Things I Meant to Post, But Didn’t Get Around to.  Except I’m not sure that there are 10, so this might be a Hitchhiker Trilogy kind of Ten.

Thing #1 was really Saturday’s post:  The fact that the subtraction principle in Roman Numerals evolved gradually and (really, like almost everything I think) with some back and forth.

And Thing #2 is this Mandelbrot video, which was passed along by a colleague (Thanks Betsey!) in October, only a few days after Benoit Mandelbrot’s death.  So in honor of the man and all that he did, here’s a tribute, prepared several years ago.  I hope he saw it and enjoyed it.

From Youtube:  “A music video for Jonathan Coulton’s song Mandelbrot Set by Pisut Wisessing made in Film 324: Cornell Summer Animation Workshop, taught by animator Lynn Tomlinson every summer for Cornell’s summer session, in the department of Theatre, Film & Dance.”

### Roman Numerals…not quite so simple

January 1, 2011

Happy New Year!  And since the New Year is all about numbers (especially if you have come to look forward to Denise’s annual January 1 post on Let’s Play Math:  form all the integers from 1 to 100 using (exactly) the digits 2, 0, 1, 1 and common mathematical symbols), here’s a picture of a number that I meant to post in October November December.

LIIII

Recognize this number?  Even though it’s not written as LIV?  This is from the 54th entryway to the Colosseum in Rome, which was built almost 2000 years ago when Roman numerals didn’t always use the subtraction property that we’re taught, where 4 is written as IV instead of IIII.

I found that to be interesting in and of itself, since I’d heard that the subtraction was a later addition but never witnessed it.  But what’s weird?  It wasn’t a sudden change.  Here’s the forty-fifth gate:

XLV

The subtraction principle was used with 40, just not with 4.  Which leads to a natural question:  what about gate 44?

XLIIII

I’m bummed that we didn’t get a better picture of this, but you can kind of see all four Is after the L.  Apparently, according to our usual font of knowledge, the reluctance to use IV is because that was the standard abbreviation for Jupiter’s name in Rome (IVPPTER), and this mixture of sometimes using four symbols in a row continued for more than a thousand years:  in the 1390 English cookbook The Forme of Cury (here on Project Gutenberg) the author still uses IIII [as in the Table of Contents, where Section IIII is rapes in potage] and there are also some IV for section numbers and references to Edward, though those might be later additions.

Published under GNU-FDL

And even 100 years ago [last year, in 1910], the Admiralty Arch in London uses MDCCCCX instead of MCMX in the inscription

ANNO : DECIMO : EDWARDI : SEPTIMI : REGIS :
: VICTORIÆ : REGINÆ : CIVES : GRATISSIMI : MDCCCCX :

(In the tenth year of King Edward VII, to Queen Victoria, from most grateful citizens, 1910).

So what does all this mean?  Nothing much, except that Roman Numeral Rules were maybe not quite as hard and fast as I once believed.

### Girth Units

December 7, 2010

This Brian Regan video isn’t new, but I saw it recently for the first time and found it hilarious (Thanks for the link Michael!).    And timely, given the holiday season.

Enjoy!

### Carnival of Mathematics #72

December 3, 2010

Welcome to the 72nd Carnival of Mathematics!  Have you been waiting all day (sorry!) for it, filled with Anticipation?  If so, that would be most appropriate, since according to this site the song Anticipation by Carly Simon was the 72nd best song of 1972.

The prime factorization of 72 is 23·32, which has a cool kind of symmetry.  Inversions also have a cool kind of symmetry, and are explored by Patrick Vennebush in Inversions « Math Jokes 4 Mathy Folks posted at Math Jokes 4 Mathy Folks.

In 1889 Nellie Bly went around the world in 72 days (a world record at the time, albeit only for a few months).  Thanks to the wonder of the internet, you can read all about it in her book.  She seems like a creative kind of gal, and might well have enjoyed the post about enclosures by Miss Nirvana in Creating Nirvana: Homeschooling: Box Assemblage posted at Creating Nirvana.

The number 72 is the sum of four consecutive primes (13+17+19+23).  It’s also the sum of six consecutive primes (5+7+11+13+17+19).  Because the primes are consecutive, the summation is pretty easy to remember.  Mnemonics also help make things easy to remember, and in Madhava’s Mnemonic Mathematics, at JOST A MON, Fëanor presents a medieval mnemonic for pi from South India.

If you want to know how fast your interest-bearing money is going to grow, you can use the Rule of 72:  dividing 72 by the annual interest rate is a pretty good estimate for how long it will take your money to double.  For example, at a 6% annual interest rate, your money would double about every 72÷6=12 years.  (This is just an estimate, and works pretty well whether the interest is compounded quarterly or daily.)    Money is one aspect that people consider when choosing a career.  Speaking of careers, Maureen Fitzsimmons presents Top 50 Blogs About Careers in Science at Masters in Clinical Research, saying, “When considering a new career, it’s always helpful to learn from people already in the field. These 50 blogs can provide that insight about science careers.”

The human body is made up of 72% water, although since I got that fact from Wikipedia I might have to retract it later.  In the post Rates of Scientific Fraud Retractions at Deep Thoughts and Silliness, Bob O’Hara explains, “OK, this is stats really – I do a quick analysis of retraction rates to see if Americans really retract more often than anyone else.  (Ha!).”

The number 72 is divisible, or nearly so, by all of the integers from 1 to 9.  In particular, it has a remainder of Two when divided by 5 or 7, and a remainder of Zero when divided by the other seven numbers, making it a bit of a Zero Hero.  For ways that you too can be a Zero Hero, see our next post, Singapore Math: 52 Ways to Become a Zero Hero by Yan Kow Cheong at Singapore Math.

World Records allow people from all across the globe to compete for bizarre bragging rights.  For example, just this past August, Patrick Lomantini set a World Record by continuously cutting hair for 72 hours in Witchita, Kansas.  A simpler way to connect to your worldwide brethren is through podcasts.   Peter Rowlett demonstrates this effectively in Math/Maths LIVE from MathsJam! at Travels in a Mathematical World, saying, “My American podcast co-host Samuel Hansen visited the UK in November and we did a mathematical tour. As part of this, you can listen to two podcast recordings made live before audiences. This is the first one, from the MathsJam recreational maths weekend.”

Another World Record was set this year by Jeff Miller of Chicago for the longest amount of time continuously watching sports TV: also 72 hours.  And another Podcast worth listening to is Math/Maths LIVE from Greenwich!, also posted by Peter Rowlett, with the note “This is the second one, from Greenwich.”

John Hart Ely, an oft-cited legal scholar, was born 72 years ago today.  It seems likely that he would be fairly well read, and so might have particularly appreciated the post The PiSBN Project by Geoff Robbins at Artificial Philosophy, which was “A personal coding project to find ISBN numbers in Pi.”

The number 72 is the smallest number whose 5th power can be written as the sum of five smaller fifth powers:
725=195 + 435 + 465 + 475 + 675
If you had to wait for an elevator when there were five unevenly spaced elevators you’d probably be happy if you’d read Where to wait for an elevator — The Endeavour by John Cook at The Endeavour.

And finally, the number 72 is 66 in Base 11.  That’s nice and straightforward.  But MarcCC at Good Math, Bad Math likes to look at arguments that are not as straightforward; his post Obfuscatory Vaccination Math (suggested for this Carnival by colleague GrrlScientist) takes a somewhat confusing argument and examines it more closely.

That’s it for this month!  Good luck to all the Putnam takers tomorrow, and the next Carnival of Mathematics will occur in January (with a Math Teachers at Play in between!)

### Sierpinski hiding in the Sistine Chapel

December 2, 2010

It was the second day in Rome, an intense day of walking and walking and WALKING, made all the harder by the youngest member of our family twisting his foot near the Colosseum.  And in a bout of bad timing, this was also the day we had tickets to the Vatican Museum (tickets that cost significantly less than 10 Billion Euros, I’m happy to say), so sore foot or not we forged ahead.

The museums were absolutely amazing, with cool things like actual Babylonian script (no idea what it means because it wasn’t clearly numbers, but still):

Plus, because it was a Friday night and the Museums aren’t always open then (last we heard it was a summer thing, extended through October), there weren’t many people in the main part of the museum.  It was dark, and we could look out from nearly empty rooms into nearly empty courtyards:

But the museums are long.  Really long.  I can’t find the dimensions, but according to my city map they look about 1/3 of a mile, and you basically walk around near the entrance then then down one whole side the entire 1/3ish mile length on the second floor, and then you go return on the bottom floor.  By the time we reached the end of the second floor we were already carrying our younger son, and we still had to walk back to get to the exit [and then walk to the Metro, and then the hotel.  And it was almost 10pm.]  But still, at this halfway point is the Sistine Chapel, and that is not to be missed, no matter how tired.

So we went in the Sistine Chapel, which was the one area that was completely crowded, plus it was really loud in there because the guards kept saying SHHHHHHHHH into the microphones and then a recorded voice came on overhead to tell everyone that this was a place of worship and to be quiet, and this was repeated loudly in 8 different languages.  So after about 15 seconds of admiring the ceiling we decided to call it a day and begin the trek back.  But then, right near the exit, TwoPi suddenly whispered, “It’s a Fractal!”  And so I looked at the floor:

See all those Sierpinski Triangles???? They go all the way to Stage 3!

The entire walk back we stopped at every souvenir stand (they’re all over the museum) and had this conversation:

“Do you have any picture of the floor of the Sistine Chapel?”
“You mean the ceiling?”
“No, the floor.”
“No, sorry.”

But then the next day we went to the Mouth of Truth (a giant face where you stick your hand in the mouth, and it gets bit off if you’re a liar), which is part of the church Santa Maria in Cosmedin.  The exit from the Mouth of Truth area goes through the church itself, and lo, there were MORE Sierpinski triangles on the floor!

Here:

and here

and smaller ones here:

and curved ones here

There were some other neat shapes, too, like these

and these, which looked just like a quilt

and these

The pieces were all laid out in sections, like…well, I really did think of quilts every time I looked at the floor:

There were even swirly parts that formed a giant infinity.

After taking 800 pictures we finally left, but a few hours later we were at the Basilica of San Clemente, which is a medieval Church built on top of a 4th century church built on top of a Temple of Mithras, and at the most modern level the floor has the same kind of design.  We sat and rested our tired feet admired it, but didn’t take any pictures because a Mass was about to start and we didn’t want to intrude.

So what was going on?  It turns out that this style of floor is called cosmatesque, and Our Friend Wikipedia describes it as:

a style of geometric decorative inlay stonework typical of Medieval Italy, and especially of Rome and its surroundings. It was used most extensively for the decoration of church floors, but was also used to decorate church walls, pulpits, and bishop’s thrones. The name derives from the Cosmati, the leading family workshop of marble craftsmen in Rome who created such geometrical decorations.

So it’s not terribly surprising that we saw three similar floors within 24 hours, even though we’d never seen anything like it before.  Sierpinski is hiding out all over the place.

### Clocks Around Rome, Part II

December 1, 2010

A few more clocks to show!

Up in the Pincian Gardens, where all the Math Guys are, is a water clock created back in 1867.  It only worked for about 40 years, however, and then was in disrepair for about a century.  Fortunately, only three years ago the clock was restored and now it totally works. Yay!  Here’s what it looked like when we were approaching it:

And here’s what it looked like when we were standing in front of it:

(Many of the clocks in Rome used Roman numerals, heh heh.)

Here’s a close up of the water portion:

The water pours first on one side, then the other.

Finally, here’s a close up of the plaque, which tells a little about it, if you read Italian:

And a 2007 article here by Brian Barrow which tells even more, including:

The timepiece is the result of the work of two men: Father Giovan Battista Embriaco, a Dominican priest and scientist (1829-1903), and the Swiss-Italian architect Gioacchino Ersoch (1815-1902). Apart from teaching physics and mathematics, Embriaco had the hobby of constructing mechanical water clocks (see box) in which the continuous emptying and filling of containers at the ends of a balanced arm produced the rocking motion which took the place of the traditional pendulum by moving a notched wheel at regular intervals.

Despite seeing quite a few neat clocks in Rome, we missed the six-hour clocks.  We’d found information about these on a site of Curious and Unusual things in Rome [a fabulous resource!], where it said:

When clocks finally began to appear on important churches and public buildings, some of them had a dial with only six hours, not twelve as in ordinary clocks, so to divide the day into canonical hours, when the prescribed prayers were to be recited. The bells, instead, rung up to twelve times, despite the dial, and the hours were counted up to 24! For instance, at the 21st hour (i.e. around 4 pm in summer) the dial would have shown III, and nine tolls of the bell would have been heard.

Only two of these dials are still extant, in the main cloister of Santo Spirito in Sassia complex, near the Vatican, and on the façade of Santa Maria dell’Orto’s church, in Trastevere district (pictures on the right).

We did sneak over the Santo Spirito, but couldn’t find the clock and there was a wedding just getting out (all the cars had big bows on the antenna; we saw this in another wedding procession the next day) so we didn’t really want to stand and look around.  I’m still not sure where it is.

BUT, as a bonus, we did unexpectedly run across two more sundials in the museum in Ephesus.

Ephesus was a Greek city before it became a Roman city before it became a Turkish city, which probably explains the Greek.  (Although it’s interesting that it’s letters instead of numbers.  Unless the letters are also numbers?  And if not, aren’t some letters missing?  I’m so confused.  Most of the stuff in the courtyard was unlabeled, so I couldn’t find out anything additional.)

Next up, even more math in Rome!  Unless I don’t get to it before Friday’s Carnival of Mathematics, in which case the Carnival will be the next up.

### Clocks around Rome, Part I

November 2, 2010

I like clocks, and in planning “How can we cram relaxingly fit many interesting things into just a few days?” I found out that there were a lot of really interesting clocks around Rome.  With very sore feet we managed to see most of them.  In chronological order (heh heh):

This is the Sundial of Augustus.  It’s an obelisk that was originally erected by Psammetichus II (aka Psamtik II) in the sixth-century BCE in the city of Heliopolis by the Nile Delta, but was taken to Rome by Augustus (aka Gaius Julius Caesar Augustus aka Octavian) in 10 BCE, where it became the gnoman (stick-thingy) if a GIGANTIC sundial.

This particular photo was taken (and placed into public domain) by someone named Arpingstone, and it’s much better than any we could have taken, particularly because it was dark when we went to this obelisk.   Which would have been a terrible shame if it were still a working sundial, but it isn’t.  I mean, it still casts a shadow, but I’m not sure if this is the spot it was originally placed on (it fell down for a few centuries); more significantly, the original lines for the sundial, which might have looked like this 19th century painting

but might have just been a meridian [marking noon], seem to be under buildings and stuff.  Rome just doesn’t look like that painting anymroe — it’s a lot more crowded.  So this is only part of a sundial, but it’s still pretty impressive.  (References:  this official sounding page and this Wikipedia article).

Jump forward about 1500 years.  The Baths of Diocletian were built about 1700 years ago and used for over 200 years; part of the remains of the frigidarium (the cold water part of the baths) were turned into Santa Maria degli Angeli e dei Martiri thanks in large part to Michelangelo.  According to Wikipedia,

At the beginning of the eighteenth century, Pope Clement XI commissioned the astronomer, mathematician, archaeologist, historian and philosopher Francesco Bianchini to build a meridian line, a sort of sundial, within the basilica. Completed in 1702, the object had a threefold purpose: the pope wanted to check the accuracy of the Gregorian reformation of the calendar, to produce a tool to exactly predict Easter, and, not least, to give Rome a meridian line as important as the one Giovanni Domenico Cassini had recently built in Bologna’s cathedral, San Petronio.

Here’s a picture from 1703 of how the whole thing would work, from Bianchini’s De nummo:

And this is how the left-hand side of that picture now looks:

See that hole in the wall in the upper right?  Here’s a close-up:

This lets the sunshine in, and there’s a cut in the cornice so that the light shines on the floor.  This is on the floor:

It’s a meridian, and I think the sunlight is supposed to strike it at noon, with “noon” referring to whatever time the sun is as high as it’s going to get that day.  But we were there around noon clock-wise and I looked for sunlight and couldn’t find it.  (In this picture, though, it almost looks like there is some light near the meridian.  It’d be really cool if that was the missing sunlight, but it might just be candles.)

Here’s a diagram that explains it all (click for a larger version).  It’s all in Italian, though.

There was more on the floor — concentric ovals which might have had something to do with Easter, and another meridian-looking thing that was raised in a display box held up by feet:

This meridian, by the way, served as an official timekeeper for about a hundred and fifty years.  After that mid-day was marked by another sundial and a cannon fired at noon from the Castel de Sant’Angelo, a tradition that is kept up even today in the form of a cannon fired at noon from the top of the Janiculum Hill.

More clocks coming up!

### Math Guys in Rome

November 1, 2010

The Villa Borghese Gardens form a giant park in Rome, and at the western edge of it are the Pincian Gardens, so named because they’re at the top of the Pincian Hill.  (Belated note to self: the fact that they were on top of a hill means it should not have been any sort of surprise that there were many many steps to get up to the Gardens.)

These were [this was?] the first public park opened in the city, and around 1850 a bunch of busts of prominent Italians were commissioned for the park.  Some of these were kept in the park, some were moved and then moved again, and some were altered to represent Italians who seemed more worthy of being commemorated.   Then through the 1950s more busts were added and there are now a total of 228, of which 225 are of men and 3 are of women.

There’s a map of all the busts online [here, along with all the history], so it was pretty easy to search out mathematicians.  Here’s Archimedes:

This was one of the original busts, but back then it was of Niccolò Machiavell; it got re-formed into Archimedes around 1860. (You might be wondering, too, at Archimedes Italian background.    A few of these busts were a little more liberal than others on what it meant to be Italian.)

This next one is of Giordano Bruno, born in 1548:

Bruno was a big fan of Copernicus’s still-unpopular view that the earth revolves around the sun, though he also thought that the sun was nothing unique either — just one of an infinite number of heavenly bodies.  Poor Bruno didn’t get along too well with the church of the time, and was burned at the stake in 1600.

On a lighter note, here’s Leonardo da Vinci, along with a rose that someone left for him:

(I just noticed the square around his face.  What’s that about?  It’s in the few other pictures that we took of him, too.)

Next up is Giuseppe Luigi Lagrangia, also known as Joseph-Louis Lagrange (though Wikipedia and Mactutor say his middle name was Lodovico originally).

He looks totally proud of everything named after him, like the Lagrangian and Lagrange Multipliers.

Here’s Pythagoras (another “Italian”) with two of his closest friends:

And finally, this is Niccolò Fontana, who became known as Tartaglia (stutterer) because the French invaded his hometown of Brescia when he was a teen and sliced his face. Ugh.

He translated Euclid into Italian and is also known for his role is finding a general solution to the cubic equation, which deserves a post all to itself someday.

And that’s it!  In theory Galileo should be in this group, but we couldn’t find him (we think he was hidden behind a construction fence), and so should Barnaba Tortolini (not sure why we missed him).  Oh, and there was also an obelisk and this really cool water clock, which was one of the main reasons that we went to this neck of the woods in the first place, but that will appear in the next post…

### Greek Math (OK, just Greek)

October 28, 2010

TwoPi and I just got back from a long-anticipated trip across the Atlantic.  And, except for the fact that camels are about fifty times taller than they look and really really scary to ride (unless you’re ten.  Then, apparently, they’re totally cool.) the trip was amazing.  Especially because we found mathy things, and who doesn’t like mathy things???

This, by the way, might be my favorite photo:

Two of the days we spent in Greece, and everything was in Greek.   Which is obvious, but it made it seem like there was math everywhere.  Even on Sprite bottles.

Plus a lot of the signs were posted in both Greek and Latin alphabets, so I could try to sound out the Greek and then see if I was right.  [I spent a similar amount of time reading signs in Montreal, once, and then checking myself on the English subtitles].

It was like all these years I’ve spent learning and teaching math symbols paid off in a completely unexpected way.  (Even though, ummm, joining a sorority might have had the same effect.  But I digress.)

Unfortunately, with only a couple days, we didn’t do anything that had any actual math content during this portion of the trip.  But I did find this sign, which made me really happy.  (I blacked out the Latin part so that you could sound it out.)

October 6, 2010

If you want to go to the Vatican Museums (home of, among other things, the Sistine Chapel) you can buy your tickets in advance online. During the checkout process they list the number of Euros you owe, but they also include a few leading zeros. Having a computer that allows room for all the digits you might need is a good thing, as anyone who remembers the whole Y2K thing realizes. So kudos to Vatican City for making sure that no transaction will be too large for the checkout system to manage. But seriously, just how much money did they expect people to spend in those online transactions???

### Luggage Math Mistake No More!

September 19, 2010

Remember that post on August 1 about the math mistake?

The mistake here is that while individual dimensions were correctly converted to centimeters (by multiplying by 2.54), the Capacity was incorrectly converted, since the 4720 cubic inches were multiplied by 2.54 (and not the cube of 2.54) to get 11988.8 cubic centimeters.

I found this error a month or so before posting it so I know it was around for a while, but I just discovered that the error has been fixed.  Behold, the new stats!

In this case, the somehow-determined capacity of 4720 cubic inches is multiplied by (2.54)³, or approximately 16.387 cubic cm per cubic inch, to get 77,346.9 cubic centimeters.  But one liter is, conveniently [although not exactly coincidentally], exactly 1000 cubic centimeters, so this translates to 77.3469 liters, which does round to 77.3 liters.  The stats for other pieces of luggage were similarly updated.

I’m so happy!  I’d like to think that the People In Charge actually read this blog, or at least the email I sent them about it, but alas I have no evidence of this.  Still, it’s nice to see the mistake corrected.

### Billions and Billions of Burgers

September 18, 2010

There’s a new restaurant in Manhattan called 4Food which is geared towards Social Networkers, according to this September 16 review on CNNMoney.com:

Customers who step into the restaurant are met by staffers ready to take orders on their iPads, a 240-foot screen featuring live Twitter feeds and Foursquare check-ins, and a menu that offers more than 96 billion customizable burger options.

This paragraph goes on to provide a link to this article in which Stacy Cowley actually explains the mathematics.  It turns out that there are some items where you can only pick 1 [1 bun among 5 choices; 1 scoop of something I don't normally think of as being on a burger --  mac n cheese and sushi are two of the options -- among 18 choices, where one of the choices is "No thanks."; and finally 1 burger among 8 choices] but for the 4  add-ons like lettuce, the 12 condiments, the 7 cheeses, and the 4 meat slices, you can have as many different kinds as you want.  This means the total number of combinations becomes

$5 \cdot 18 \cdot 8 \cdot 2^{\left( 4+12+7+4\right)}$

which is 96,636,764,160.

The article contains more specific information about each of the choices, and the many many comments include additional information, like that someone in theory should be able to order no bun or no meat, although the comments mostly seem to be arguing whether or not the math is correct (no math mistakes that I see!).  Incidentally, if you allow the option not to have a bun or not to have the patty at all, which seem reasonable options to include, the 5 in the equation changes to a 6 and the 8 to a 9, giving 130,459,631,616 combinations. And that’s a lot of choice.

September 2, 2010

We finally published the Winter Spring SUMMER(ish) edition of our department newsletter!  This issue is named the Taniyama Times after Yutaka Taniyama (谷山豊 ) of the Taniyama-Shimura conjecture (proved by Andrew Wiles, and giving Fermat’s Last theorem as a wee little corollary).

This issue will admittedly hold less interest for non-alumni than most of our issues, since it’s primarily about where the Class of 2009 has been spending the past year.  Still, it does contain the following fun problems to work on!

Problem 4.2.1: (2006 AMC10) If xy =x³—y, what is h◊(h◊h)?

Problem 4.2.2: Which fits better: a square peg in a round hole or a round peg in a square hole?

Problem 4.2.3: The figure below shows the first three circles in an infinite sequence. What is the total area of the circles? What is the total circumference?

Answers are welcome in the comments, and you might just be acknowledged in the next newsletter!  [If we remember, which is sort of a risk since I'm right now remembering that we forgot to check that when we put together this issue.]