Monthly Archives: March 2013

math musings

The Pop Quiz Paradox

This is one of my all-time favorite paradoxes:

On Friday, a teacher announces to his class that there will be a pop quiz one day during the following week. In order to uphold the integrity of the quiz, it must satisfy the following two conditions:

  1. The quiz will be handed out at the beginning of class one day the following week (Monday through Friday).
  2. The students will not be able to (logically) deduce which day the quiz will be held before they are actually given the quiz.

After making this announcement, he notices a lot of murmuring around the class, and after a minute a few students start sniggering. He asks the students what they find so amusing. One of the cleverer students in the class raises her hand and explains:

“You cannot possibly give us a pop quiz next week,” she says.

“Why not?” the teacher asks.

“Well, if we won’t be able to deduce which day the quiz is before we are given the quiz, it can’t be on Friday. For if after class on Thursday we still haven’t gotten the quiz, we will know that the quiz must be the following day. This contradicts the second requirement for a pop quiz.”

“That makes sense,” he agrees.

“So the quiz can only be given one of the days from Monday through Thursday. By the same argument as before, the quiz can’t be on Thursday: Since we’ve shown it can’t be held on Friday, we will know by Wednesday if the exam is to be held on Thursday. By the same token, the quiz can’t be held on Wednesday or Tuesday. The only possible day you could give the quiz is Monday. But I know now that the pop quiz must be on Monday, which again contradicts the second condition for pop quizzes. So you can’t give us a pop quiz next week!”

The teacher tells the class that he doesn’t see any flaw in the student’s argument, so it appears that he can’t satisfy the requirements for a pop quiz. The students seem pleased to have a weekend free of study having deduced that a pop quiz is an impossibility. Imagine the students’ surprise when the teacher hands out the pop quiz on Tuesday at the beginning of class, thus fulfilling the previous week’s proclamation!

Commentary

I first encountered this paradox in a collection of Martin Gardner‘s writings entitled The Colossal Book of Mathematics, where it is referred to as “The unexpected hanging.” A more detailed account of this paradox can be found in this article by Timothy Chow. The subtlety of the paradox is suggested by the sheer number of references cited in Chow’s essay.

adventures

Joshua Tree

Yesterday I got back from a whirlwind tour of Joshua Tree National Park with my friends Sam, Humberto and Anand. We were only there for twenty four hours, but saw a fairly large cross section of the park. On Wednesday we set up camp in Black Rock Canyon and hiked up Black Rock Canyon to the panorama loop trail. The high point of the trail straddles a ridge which gave spectacular views of the Mojave desert to the north and Coachella valley to the south.

The following morning we packed up camp and drove into the heart of the park. We (mostly Sam and Anand) did a bit of scrambling on the rock formations, then drove up to Keys View to bask in the arid and rugged scenery before returning home.

Although our visit was brief, I really enjoyed my time in Joshua Tree. Considering it is only a two-and-a-half hour drive from LA, it is embarrassing that it has taken me this long to go out there. Anyway, it was well worth the visit.

  • My new tent's first outing
  • A dead tree looms above the Mojave
  • San Jacinto Mountains to the south
  • A pretty spectacular Joshua tree on the trail
  • The camp at sunset
  • Sam and Anand atop some rocks
  • A forest of Joshua trees
  • Tree, rock, and sky
  • Another spectacular horizon and sky
  • Keys View of the Coachella Valley
adventures

Desert/Modernism

Last weekend, I went with Alivia, my dad, and stepmom to Palm Springs. Other than getting a chance to relax and spend time with my family, the highlights for me were a hike through Tahquitz canyon and a driving tour of some modernist houses.

In the mid twentieth century, Palm Springs was a veritable playground for architects pioneering the modernist style. The somewhat austere minimalism of their work suits the desert landscape well. I had seen pictures and videos of many of the houses we visited before, but it was fantastic to see the buildings in situ. We saw E. Stewart Williams‘ Edris House and Sinatra House, Albert Frey‘s Frey II House and Tramway Gas Station, and Richard Neutra‘s Miller House and Kaufmann Desert House.

 

 

  • Tahquitz Canyon Wall
  • Tahquitz Canyon Rock
  • Tahquitz Canyon Floor
  • Tahquitz Canyon Falls
  • Edris House
  • Kaufmann House
  • Sinatra House
teaching

Math 32B Sample Questions


I’ve finished writing up some sample questions for the Math 32B final. The questions are available here and their solutions here. These questions are are my best guess of the sort of material that I feel is likely to appear on a final exam. As usual, feel free to let me know in the comments below if you find any errors.

teaching

Math 32B Final Review

I have written up a whirlwind review of the material covered in Math 32B this quarter. It is available here. The review is not meant to be entirely comprehensive, but I think it covers the bulk of the computational material from the course. The review is longer than I was hoping (6 pages) but it is certainly more condensed than the text which will hopefully be appealing to some students. As usual, let me know in the comments if you find errors or if anything in the text is unclear. In the next couple days, I will write up some sample problems and solutions for the exam. So stay tuned.

edutainment math

Dimensions Video Series

In this month’s issue of the American Mathematical Monthly, there is a review of a series of animated videos about geometry, called Dimensions. The aim of the videos is to introduce the geometry of 4-dimensional space, and in particular the 3-sphere which we can view as living in 4-dimensional space. I haven’t finished watching the entire series, but what I’ve seen so far is pretty impressive. The entire series runs about two hours. It is freely available on youtube — see below for the first installment.

adventures

Hiking in Point Mugu State Park

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On Saturday, my friend Peter and I went for a hike in Point Mugu State Park. We intended to follow the trails written up here, but took a wrong turn. So we ended up making our own route.

The park is situated at the western end of the Santa Monica mountains, where they descend into the Pacific Ocean. I was astonished by the diversity of the landscape and flora there. We entered the park from north, where the scenery was dominated by rolling hills and grassland. Less than a mile into the park, we started ascending into the brush and succulent covered mountains that I so strongly associate with Southern California. Our trail led us into the forested Sycamore Canyon, terminating at a refreshingly cool albeit anemic waterfall.

We descended back along the stream bed down the canyon until it met up with Fossil Trail. We followed this deserted trail up an arduous ascent and were greeted with views of a network of canyons below, Boney Mountain summit above, and the Pacific Ocean to the west. True to the trail’s name, there was a rock formation containing dozens of fossilized shellfish midway to the top.

At the top of the trail, we followed another trail back along the ridge of Sycamore Canyon which offered a more gentle descent back to the valley floor. A large section of this trail was covered by a dense grove of trees arching over the path giving the impression of walking through a long narrow cathedral. Combined with the singing birds and an absence of other people, it made for a supremely serene ramble.

Peter and I only explored a small corner of the park on our hike. Given more time, I would love to return to scale the higher peaks and hike down to the ocean. I will certainly return to Point Mugu state park.

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teaching

Homework 8 Hints


I’ve had a number of requests for hints and/or solutions to several of the homework problems for tomorrow. So here we go.

17.5 #30 This question asks us to compute the gravitational flux through a cylinder \(S\) from the gravitational field
\[
\mathbf{F} = – G m \frac{\mathbf{e}_r}{r^2}.
\]
Here, \(S\) is the cylinder of radius \(R\) whose axis of symmetry is the \(z\)-axis with \(a \leq z \leq b\), and \(r = \sqrt{x^2 + y^2 + z^2}\) is the distance from \((x, y, z)\) to the origin.

Recall that \(\mathbf{e}_r(x, y, z)\) is the unit vector that points in the direction \(\langle x, y, z \rangle\). Therefore, we can write
\[
\mathbf{F}(x, y, z) = – G m \frac{\langle x, y, z \rangle}{(x^2 + y^2 + z^2)^{3/2}}.
\]
In general, if \(H(u, v)\) is a parametrization of a surface \(S\), we compute the flux of \(\mathbf{F}\) through \(S\) to be
\[
\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D F(H(u, v)) \cdot \mathbf{n}(u, v)\, du \, dv
\]
where
\[
\mathbf{n}(u, v) = \frac{\partial H}{\partial u} \times \frac{\partial H}{\partial v}
\]
is the normal vector for the surface \(S\). For this problem, we cannot parametrize the cylinder with a single equation: we must parametrize the top, sides, and bottom separately. Hence, we must compute 3 integrals to determine the flux.

The sides of the cylinder can be parametrized in cylindrical coordinates by
\[
H(\theta, z) = (R \cos \theta, R \sin \theta, z)
\]
where \(0 \leq \theta \leq 2 \pi\) and \(a \leq z \leq b\). You should check that this parametrization gives a normal vector \(\mathbf{n}(\theta, z) = \langle (R \cos \theta, R \sin \theta, 0 \rangle\). Therefore, the contribution of the flux from the sides of the cylinder is
\[
\iint_{\mathrm{sides}} \mathbf{F} \cdot d\mathbf{S} = – G m\int_a^b \int_0^{2 \pi} \frac{R^2}{(R^2 + z^2)^{3/2}}\, d\theta\, dz.
\]
You can compute the outer integral using the trig substitution \(z = R \tan u\).

For the top of the cylinder, we must parametrize the disk of radius \(R\) parallel to the \(xy\)–plane centered at \((0, 0, b)\). Again, we can use polar coordinates to parametrize this region as
\[
H(r, \theta) = (r \cos \theta, r \sin \theta, b)
\]
for \(0 \leq r \leq R\) and \(0 \leq \theta \leq 2 \pi\). This parametrization gives a normal vector of \(\mathbf{n}(r, \theta) = \langle 0, 0, r \rangle\). Notice that this is the upward pointing normal vector, which is what we want since we need an outward pointing normal for the entire cylinder \(S\). Therefore, the integral for the flux through the top face of the cylinder is
\[
\iint_{\mathrm{top}} = – G m \int_0^R \int_0^{2 \pi} \frac{b r}{(b^2 + r^2)^{3/2}}\, d\theta \, dr.
\]
This integral is easily computed using a \(u\) substitution.

The flux through the bottom face of the cylinder is almost identical to the top face except that \(z = a\) on the bottom face, and we must take the downward facing normal vector \(\mathbf{n}(r, \theta) = \langle 0, 0, -r \rangle\). When computing computing the integral for the bottom face, you will get a term involving \(\sqrt{a^2}\). Be careful that you simplify \(\sqrt{a^2} = |a|\), and not just \(a\), for this will make a difference (in fact the crucial difference) when \(a\) is negative instead of positive.

18.1 #40 This problem asks you to show that when \(\mathbf{F} = \nabla \phi\), we have \(\mathrm{curl}_z(\mathbf{F}^*) = \Delta \phi\). Recall that for \(\mathbf{F} = \langle F_1, F_2 \rangle\)

  • \(\mathbf{F}^* = \langle – F_2, F_1 \rangle\)
  • \(\mathrm{curl}_z(\mathbf{F}) = \frac{\partial F_2}{\partial x} – \frac{\partial F_1}{\partial y}\)
  • \(\Delta \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2}\).

Using these definitions and \(\mathbf{F} = \nabla \phi = \langle \partial \phi / \partial x, \partial \phi / \partial y\rangle\), we can compute
\[
\mathrm{curl}_z(\mathbf{F}^*) = \mathrm{curl}_z( \langle – \partial \phi / \partial y, \partial \phi / \partial x \rangle) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2},
\]
which is exactly what we wanted to show.

art

Kinetic Sculptures by Grönland and Nisunen

Tommi Grönland and Petteri Nisunen are artists from Finland. They specialize in kinetic sculpture. I found this video of one of their sculpture “Unstable Matter” particularly mesmerizing:

I love the patterns created by the ball bearings both in motion and stationary, and the way your perception of the motion is completely different at different scales. In the zoomed–in view you notice the individual bearings with their seemingly predictable motion. On a larger scale, you notice the seemingly random patchwork of grid–like patterns created by the bearings. Finally on the largest scale the motion appears continuous and wave–like. It reminds me of some patterns that appear in the Ising model of magnetism. The net effect is entrancing — I would love to see one of these sculptures in real life!

Grönland and Nisunen have more videos of their work posted here.