Carnival of Mathematics #48

You blink your eyes and all of a sudden two weeks have passed, and it’s time for another Carnival of Mathematics!  This one, the 48th edition, is being hosted by the group blog Concrete Nonsense.   As usual, there is a whole bunch of great stuff there, from models of how long people spend in coffee shops to integrals, cabbages, music, and the US Constitution.

(Concrete Nonsense also has a recent post I particularly enjoyed, called “Wood, glue, and the Octahedral Axiom”.  It’s not my area of math, the post and looking at the model were still really interesting!)

Happy reading!

[Incidentally, we’ll be hosting the next Carnival, on Friday the 13th!]

Simple Addition: Apparently not so simple

Poor Montgomery County, Maryland.  Someone forgot to check the addition on the estimate of property value in the county, and it turns out that the estimate wasn’t a very good estimate after all.  It was off by a whopping $16,000,000,000.  That’s right, sixteen billion dollars — a result of entering $180 billion instead of $164 billion.  To put that kind of money into perspective, it’s….it’s sixteen billion dollars.  There is no way to put that in perspective.   [Except perhaps for  The Washington Post’s observation that it is equivalent to the Gross Domestic Product of Jordan.]

Of course, that’s just the value of the property,  not the amount of revenue that the government expected to gain in taxes from the property value, which was used to calculate school budgets.  Ironically, the result was that Montgomery County didn’t get enough money:  money from richer counties is shared with schools in poorer counties, and Montgomery looked extra rich.  The end result is that Montgomery is getting its $24 million, but lawmakers are suddenly trying to find a way to cover the $31 million that was mistakenly given to the other schools from Montgomery’s non-existent tax income.

(“Suddenly?” you ask.  “Didn’t this mistake actually happen in 2007?”  Why yes, yes  it did, in November.  A few people noticed:  one person even sent an email asking if the numbers were correct, but the email was never answered.  Having forgotten to reply to a few emails myself, I can totally see how that happened — you get a question, mean to answer it, get distracted, and all of a sudden it’s fallen down the email queue into The Abyss.  A bit of a problem, nonetheless.  So the matter was tabled, until this past summer when “mid-level number crunchers in state government” (WP) caught it — hooray for the mid-level number crunchers!   Word got back to Montgomery County last month, and this story first hit the airwaves a few weeks ago.)

The moral of the story:  check your math.  And check your email.

Bochner’s Meditation on the Theorem of Pythagoras

The January 2009 issue of The College Mathematics Journal has a Pythagorean theme.  While the articles consist of the usual mix of varied mathematical topics, most of the smaller sidebar inserts contain quotes from books or articles about Pythagoras, and the issue concludes with reviews of recent books by Eli Maor and by Christoph Riedweg on the Pythagorean Theorem and the life of Pythagoras, respectively.

The front cover of the journal has a photograph of a piece created by the artist Mel Bochner, his “response to a visit to a temple in Metapontum”, the city where Pythagoras is said to have died.  Media:  chalk and hazelnuts.  (You can also find a different image of this work, dating to 1972, as the 16th image in the slide show of “Selected Works: 1966 – 2008” on Bochner’s website.)

I love the simplicity:  illustrating the fundamental ideas of relating the lengths of the sides of a triangle to the areas of the squares on those sides using readily available materials.

I do have one nit to pick, though.  If the intent was to illustrate a right triangle, then the arrangement of hazelnuts is off.  To my eye, the hazelnut grids look exactly like pins on a Geoboard, or lattice points in the plane.  And given that perspective on this image, we see a 2-3-4 triangle, an obtuse triangle, and squares of area 4, 9, and 16.

I suspect that what was intended was something akin to the following:

Here we can view the diameter of each hazelnut as being our unit of length, so that the circular area taken up by each hazelnut suggests the unit of area (the circumscribing square) .

This image differs from Bochner’s piece in a critical way:  Bochner has arranged his hazelnuts with relatively large gaps between each nut, while in the schematic I’ve abutted them to one another, as one would do if the diameter of each nut was a unit of length, and the nuts were being used as a measurement device.

The large amount of space between the nuts is akin to lattice points in the plane, in which it is the gap itself which constitutes the unit of length, and the vertices (or hazelnuts) are our attempt at an approximation to ideal points in the plane.

If the triangle is meant to be a 3-4-5 triangle, the corresponding lattice image would be as follows:

In the end, I find Bochner’s Meditation rather confusing, and to some extent disappointing.

What’s the (Geometric) Pattern?

To go along with yesterday’s post, here is another fun “Find the Pattern” problem.  I got this from Helen Timberlake, but I’m not sure if it was her creation or not.

What’s the pattern?

1, 11, 21, 1211, 111221, 312211, 13112221, …

This sequence was the favorite pattern of my former department chair, Nelson Rich.  I thought he invented it, but a quick search on the internet reveals that it’s pretty easy to find if you search for the first few terms.

But don’t do that! It’s a sneaky one, but fun to try and figure out.

The Night Before Comps

Our seniors are taking their comprehensive exams, a department requirement for graduation, this weekend.  They’ve been studying for months (literally — we gave them the review packets last April and I know several people were working on them this summer) and now The Time Has Come.

Last week one of our seniors sent me this poem that she’d written in honor of the occasion, and she gave me permission to post it here.

Good luck everyone!

The Night Before Comps
A parody of Clement Clark Moore’s The Night Before Christmas
By Elyse Matson

’Twas the night before comps and all through Nazareth College,
The math majors were studying, stuffing their brains full of knowledge.
Notebooks and mechanical pencils scattered the room
In hopes that their proofs had the right “we assume”.
Coffee mugs nestled snug in their grips
With visions of scoring at least a 76.
With students in the library, and in the math center
Settling down for an all-nighter
When at one table there arose such a clatter,
The math majors looked up to see what was the matter.
Over to the commotion they went with a flash,
Someone was crying, sure she wasn’t going to pass.
The eigenvectors and values were giving her trouble
And she needed help with derivatives on the double.
When, what to the math majors wondering eyes did emerge,
A Solutions manual complete with p-series that diverge!
With step-by-step directions, and answers too
They knew in a moment they were going to pull through.
No more would they suffer from their torments,
They cheered and shouted and read the table of contents.
“On normal distributions and integration by parts,
On epsilon delta, and z-score charts.
To contrapositives! To proofs by induction!
Now integrate, integrate using u-substitution!”
As the students sat down and started to study,
The formulas made sense; they weren’t so fuzzy!
So question after question their mechanical pencils flew
Through linear, statistics, calc one and calc two.
Next the students began writing their proofs,
With enough “hence’s” and  “thusly’s” for their professors to approve.
After they took limits as n approaches infinity,
They stretched their legs and refilled their coffee.
Although sleepy and very tired,
They studied and studied doing all the questions required.
Their eyes – how they drooped! Their faces not so merry!
Their minds were fried, their stomachs empty.
Their problems complete, formulas whizzing through their heads,
The math majors went home to their beds.
And when they awoke to take their exam,
One of them uttered a not so quiet, “damn!”
“What’s wrong?” they asked, with concerned looks.
“We didn’t find out who sent us that book!”
The math majors asked around the college,
But couldn’t find a person to acknowledge.
After the exams were done and scored,
The students relaxed; they were glad to be bored.
And although, they never found out who helped them that night,
They all passed and got every question right!

Pick 3 Coincidences

Nebraska has a lottery.  One of the games is “Pick 3”, where each player selects 3 one-digit numbers, and once a day a computer selects a winning sequence.  The player wins if they have the same three numbers in the same order.

Monday (Jan 19, 2009), the winning numbers were 1-9-6.  One lucky Nebraskan had bet the same combination, winning the top prize of $600.

Tuesday (Jan 20, 2009), the winning numbers were ALSO 1-9-6.   This time, three people had the winning combination.  (No, the person who won on Monday didn’t play the same combination on Tuesday.)

The news accounts of this event say that the two drawings were done on different computers (preempting the unspoken suspicion that there was some systematic error in the process).

According to the Associated Press, the probability of this happening are one in a million.  And indeed, there are 1,000,000 possible combinations of winning numbers over two consecutive nights, since if you just concatenate the digits you get all the strings from 000000 to 999999.

But to say the probability is one in one million means you’re measuring the probability of having 1-9-6 come up a winner in both drawings.  Most readers of the news story probably aren’t struck by that specific combination, but merely from the fact that the two numbers agree.  That sort of coincidence is far more likely to occur, with a probability of 1 in 1000 (since there are 1000 possible draws on Tuesday, and only 1 of them will match Monday’s combination).

The probability that two consecutive draws do not match is 999/1000.  But the probability that after n+1 drawings, no consecutive pair will match, is .   For n=366, we find that the probability of no matching pairs to be roughly 70%, which means that over the course of one year of Pick 3 drawings, there’s a 30% chance of having the same numbers win two nights in a row at least once that year.

In two years of Pick 3 drawings, there is a more than 50% likelihood that the winning numbers will match on some pair of consecutive evenings.

As one of my Philosophy professors used to say, “Some surprises are not unexpected.”

Yet more math in bones: calculating weight

Yes, more math in Bones!  This one from “The Truth in the Lye”, Season 2 Episode 5.  In this episode, a body was found in a bathtub that had been filled with lye, so the body was in an advanced state of decomposition.  (My apologies if you’re reading this over a meal.  It only gets more descriptive, so you might want to skip this post.)

Here Camille Saroyen is talking to the team at the Jeffersonian about how much the person would have weighed.  (I switched to first names for the dialogue, because I had no idea what Camille’s last name was until I looked it up.  Speaking of looking this up, after trying to transcribe it from the DVD I discovered that the whole thing had been transcribed for me here).

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

Camille: What’s our starting weight, Zack?

Zack: Starting weight is 542.13. [lbs]

“Bones”: The tub itself weighs about 200 pounds. Capacity is 34 gallons.

Camille: Which at about 8.3 pounds a gallon comes to 270-275.

Jack: And two-thirds full makes it about 180, putting this guy somewhere in the 160-pound weight class.

[Brennan nods]

[Cam is stirring the tub, where orange is starting to appear]

Camille:The cream always rises. Or in this case, melted body fat. [raises tong, melted body fat drips off] I’ll measure its volume to determine body type.

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

My initial thought was to figure out if this was good science or not, assuming that it must be because, hey, these people work at the Jeffersonian.  I couldn’t find the weight of a bathtub to check their very first calculation exactly; however, the estimate I found here suggested that clawfoot tubs were 250-350lbs, and modern tubs weigh less, so that 200lbs might well be in the ballpark.

But I started to worry about all the rounding.  Does the tub weigh 200 lbs exactly, or could that be off by 10 lbs?  Then there’s that whole “about two-thirds full”.  Close, but again I suspect a wide margin of error, especially since they rounded when they said that 34 gallons times 8.3 pounds/gallon was 270-275 (It’s 282.2, which is  10 pounds off in a situation that implies accuracy to more than 10 pounds.)

All in all, I think their final answer of 160 lbs (from 542.13-200-180, which equals 162.13) is really pretty rough.  Ballpark, maybe, but it could reasonably be anywhere from 140 to 180 lbs or more.  I want to believe in their good mathematics, but I fear that Barry Leiba was right when he implied in his comment earlier that they’re not really trying to show good science.*

*Not that that will stop me from continuing to talk about it.

My Presidential Age

I celebrated a Presidential Age for the last half of 2008.    By that, I mean I was 43 years old during the term in office of the USA’s 43rd President.   And in fact, later this coming year I will turn 44, and have a second Presidential Age.

In this post, I want to explore some of the properties of this notion of “Presidential Age”.  In particular:

• Everyone who lives long enough should expect to experience a Presidential Age of their own,
• only some people are likely to experience two, and
• experiencing three or more is rare but possible, and has happened in our nation’s history.

In what follows, I want to prove the first statement, explore the likelihood of the second, and give examples of the third.

Everyone who lives long enough will experience a Presidential Age.

Assumptions:  A)  People live arbitrarily long lives, and B) over the course of your life, the mean length of a Presidential term in office will be greater than 1 year.

Roughly speaking, the reason why everyone gets a PA is the fact that your age increases by 1 each year, while the President Number increases more slowly than that on average, so eventually your age must become the larger of the two quantities.  (Granted, in a few hundred years, the Presidential numbers will be larger than 100, and it will be harder for people to live long enough to experience their PA, but the general principle still holds.)

Making the previous paragraph more rigorous is at first glance non-trivial, as the two functions involved (your age and the Presidential number) are not continuous, so the Intermediate Value Theorem from the calculus doesn’t apply.  And in fact you can have  rational valued functions where f(a) < g(a) for all small enough a, and f(b) > g(b) for all large b, without f(x) = g(x) at any intervening point.  (See the comments to a puzzle on winning percentages from jd2718’s blog for one such example.) 

For the Age and Prez functions, though, it is significant that they are integer-valued, their values only increase, and they always increase by exactly 1.  Thus the only way my Age can first  become strictly larger than the Prez number is if I have a birthday (so my Age jumps by 1) on a day when there is no change in administration (so the Prez number stays unchanged), and hence just prior to my birthday my Age must have actually matched the Prez number.

Getting more than one Presidential Age

Clearly lots of people will have more than one Presidential Age.  Anyone who turned 43 between January 21, 2007 and January 19, 2009 will have been 43 years old during the GWB era, and will be 44 years old during the BHO era.  But those who turned 43 earlier during the Bush presidency will have missed their chance, having turned 45 before President Obama took the oath of office.

The likelihood of getting a second Presidential Age depends, then, on the length of the term of the relevant President.  If all Presidents served their full term in office, then in order to get more than one Presidential Age, your Age and Prez numbers must first coincide within two years of the end of that President’s term, and we’d expect this to occur between 1/2 and 1/4 of the time (depending on whether that president served one or two terms).

Reality is of course not so simple as all that, as some Presidents served less than their full term in office.  Some examples:

Nixon-Ford-Carter

Richard Nixon was our nation’s 37th President.  He resigned the presidency on August 9, 1974, and was succeeded by Gerald Ford.  Jimmy Carter followed as the 39th President on January 20, 1977.

An individual whose was born between 1/21/37 and 8/9/37 would have been 37 during the last months of the Nixon administration, 38 during Ford’s, and 39 at the beginning of Carter’s term in office.

Van Buren – Harrison – Tyler

A more extreme example occurs in the 1840s, with the death of William Henry Harrison just one month after taking office.  In that case, anyone who was nine years old between March 4, 1841 and April 4, 1841 would have three Presidential Ages (8, 9, 10).

More than three?

In principle, it is possible to have more than three Presidential Ages, although for that to happen would require  consecutive abbreviated terms in office, something our nation has never experienced, and hopefully never will.

Carnival of Math #47: The Star Trek Edition

The 47th Carnival of Mathematics is up today, courtesy of regular host jd2718.  The number 47 appears a lot in Star Trek (particularly Star Trek:  The Next Generation), so this Carnival is written with a Star Trek theme in mind.    Now I’m torn between reading more of the math posts and reminiscing about ST:TNG, which I used to watch every week in college.  But fortunately the weekend is only half over, so there is plenty of time for both.   Enjoy!

The photo of George Takei is by Zesmerelda, licensed here under Creative Commons.  And OK, he’s in a parade, but that’s LIKE a Carnival.

Edited 1/18 to add: This post was actually written by Ξ, not TwoPi — I wrote it before realizing that TwoPi was logged in, and am not sure how to change the authorship.  Whoops!  And this wouldn’t be a big deal, of course, except that TwoPi is a fan of the original Star Trek, not so much TNG.

Quick: what’s 500+500?

We’ve seen Joey discuss the value of long division (when wanting to portray someone who has just gotten bad news);  now enjoy a Friends excerpt where his addition skills get a work-out:

(You can find the entire scene here.)

Watching this video clip, I’m struck by a number of mutually tangential thoughts:

• I see a LOT of my students pull out their calculators to do basic arithmetic these days (multiplying a two digit number by 2 or 3, or adding two 2-digit numbers).  I’m often surprised during a calculus quiz to see someone use a calculator to check their integer arithmetic (in combining like terms of an algebraic equation, say).
• I’m heartened to hear laughter in the video, and wonder just how long it will be before American audiences see nothing uncomfortable or amusing in an adult needing a calculator to find 500+500.
• I’m reminded that in laboratory studies of human responses to stress, one of the standard ways to ethically induce stress on test subjects is to have them perform multidigit subtraction in their head (e.g. count down from 483 by sevens) while the experimenter urges them to work more quickly.

But most of all, I’m reminded of how much I miss watching television, and wonder how I ever used to find the time to do it.  (Oh yeah, that’s right, that was life before parenthood.)

Thank you to Ionica at wiskundemeisjes, who brought this Friends scene to our attention, and suggested that we both post about it on our blogs simultaneously!

Temps fail the Intermediate Value Theorem

Rochester’s in the middle of a cold spell right now, with highs in the single digits.   And apparently sudden jumps in temperature.  Check out today’s paper:

On which day are they predicting that it will be 11°F  (or -12°C)?

Silly applications of factoring quadratic polynomials

One of the nicer talks that I attended at the Joint Mathematics Meetings in DC was given by Jeff Suzuki, on “A History of College Algebra in the United States During the Nineteenth Century“.   Suzuki’s talk focused on equation-solving, and he noted that factoring hadn’t been a significant equation-solving tool throughout that era.  Equations were solved by other means (e.g., for quadratics:  completing the square, or the quadratic formula;  for higher order equations, roots might be found/estimated by bisection or other iterative approximation schemes).

This reminded me of one of my pet peeves from teaching algebra.  Modern elementary algebra texts teach four ways of solving quadratic equations:  graphing (and finding x-intercepts by inspection), factoring, completing the square, and the quadratic formula.  Most current books aimed at the “college algebra” market emphasize factoring.  (One of my daily reads, jd2718, has an interesting take on the role of factoring in HS algebra.)

What irked me when last I taught college algebra were the “applications” of quadratic equations.  (Too) Many of the applications amounted to numerical exercises:  “Find two numbers whose product is 45 and whose difference is 4” was a typical example.

The intent is that a student will introduce one or two variables, representing the two numbers as either x and x+4 or as x and y, and eventually arrive at the equation , either directly or via a system of equations.

From here, we solve the related equation , by finding two numbers whose product is -45, and which add to +4:  that is, two positive integers whose product is 45 and whose difference is 4.

The usual technique for achieving this is to list all integer products which equal 45, and find the factorization by inspection.  But of course this amounts to a direct solution of the original word problem, without recourse to any algebra whatsoever.

Granted, there are some advantages to this algebraic method, which become clear when you consider situations with non-integer solutions.  For example, if we want to find two numbers whose sum is 6 and product is 10, trying a brute force attack is unlikely to work.  Setting up the problem algebraically, and reducing it to solving , allows us to apply another paradigm — completing the square — to eventually find the two numbers and .

I suppose that there just might be some slight pedagogical value in having students see that solving their quadratic equation is equivalent — literally and explicitly — to solving the original word problem.  But this amount of circularity always struck me as a bit daft, and I feared the day when an eye-rolling student in the back of the class was going to point out that the Emperor was underdressed.

Blue Man Math Contest

The Blue Men want you — yes, YOU — to use your mathematics skills for a contest! Well, if you’re a high school student that is.  They are inviting “high school science, performing arts, and math students” to enter a contest to create a musical instrument out of unusual items (for inspiration, think of the PVC pipes that they use in their performances).

But act quick — the deadline is January 15!  The official stuff is here.

Speaking of the Blue Man group, did you know that they opened an elementary school in New York City this past fall?  It’s called the Blue School and the classroom reminds me just a bit of one of their performances, although the $27K price tag is a bit out of my range (though apparently standard for NYC private schools).

Blue Man photo published here by Stefan-Xp under the GNU Free Documentation License.

Polygons in the Smithsonian

While we were in DC, we managed to sneak in a visit to the Museum of Natural History.  And right next to the Hope Diamond (which, umm, looked surprisingly small.  I’d envisioned something fist-sized, which goes to show how little I know about diamonds) there were some really cool rocks.

The first thing I saw were balalt columns.

I got really excited because I wrote about last April and how they often form hexagons.  So I took a closer look:

Yup, a perfect hexagon pentagon.  But there were a few hexagons around it, and it was pretty neat looking.

Then I moved to the minerals.  There was some neat symmetry in these twin crystals:

(Here’s the info on them.)

Then I saw cubes.  Lots and lots of cubes, because a bunch of crystals grow that way.  There was this Fluorite from Spur Mountain mine, Cave in Rock in Illinois.

And these cubes of fluorapophyllite from Poona, Maharashtra in India.They were clearer in real life.

Then there’s this fluorite in gypsum, which is neat both because the cube is embedded in a see-through rock and because it’s from Penfield, which is just down the street from Rochester.  Seriously, you could pretty much walk there, and it’s an easy bike ride.

Does anyone know if these rocks are related to the stuff that’s in toothpaste?

If shiny is more to your liking, here’s a whole bunch of Galite cubes from Missouri.

Here’s the sign.  I like taking pictures of signs.  Otherwise, how would you know that there’s sphalerite mixed in?

And then check out this Pyrite.  It grows in two ways:  cubes and dodecahedrons, and this photo shows examples of both.  I love pyrite. 

I can’t wait to go back to the Smithsonian again.