Carnival of Mathematics #59

Welcome to the Carnival of Mathematics! We’ve got a full roster this time, including roller coasters, topological Turán theory, a mathematician arrested as a spy, a plane running out of fuel mid-flight, speed limits in Conway’s Game of Life, and much, much more. But first …

A polygon or polyhedron is cumulated by replacing all the edges by triangles or faces by pyramids. This toy demonstrates switching between a cube to the cumulation of a tetrahedron (also known as a triakis tetrahedron):

A related construction is a stellation, which occurs when the edges of a polygon or faces of a polyhedron are extended until they intersect. Here is a pentagon stellated into a pentagram:

In this case the stellation could be made with a cumulation instead. Some but not all stellations are also cumulations.

When enumerating stellations particular rules are generally followed. The most famous set is known as “Miller’s rules”, which in short restricts stellations to: a.) ones with a certain sort of symmetry b.) ones that form no “hidden holes” on the inside of the polyhedron and c.) ones where all parts of the polyhedron are connected.

When applying these rules to an icosahedron (Platonic solid with 20 faces) …

… there are 59 possible stellations (including the icosahedron itself), as shown in the fantastic image above. By sheer coincidence, counting the multi-part posts and this post itself, we have 59 entries for the carnival. On to the festivities!

The American Institute of Mathematics recently announced all the congruent numbers up to 1 trillion have been enumerated. But what are congruent numbers? Brian at bit-player gives them a thorough treatment.

Here’s a puzzle courtesy Daniel Colquitt involving an ant on a rubber rope.

Speaking of puzzles, Tom Lovering wants you to participate in a problem solving experiment involving an infinite hallway.

The Law of the Excluded Middle states that if a mathematical proposition is not true, it’s false. Mathematicians have experimented with dropping this requirement; Ben Burgis considers the ramifications for probability.

At Tanya Khovanova’s blog, be sure to try what I believe to be the best “odd one out” puzzle ever written.

The 4-color-theorem states that any map can be colored with at most four colors so no two adjacent regions share a color. It was proven in 1976 by Kenneth Appel and Wolfgang Haken using a computer to check 1,936 maps. Noah Snyder at the Secret Blogging Seminar almost but doesn’t quite prove the 4-color-theorem in a single blog post.

At Mathing… there’s an interesting quiz of 18 questions involving the number zero.

Conway’s Game of Life has gliders” and “spaceships” which move around the cellular automata grid (see the picture above). Nathaniel Johnston works out if they have “speed limits”.

Rich Beveridge puzzles over (sans calculus) the area under a parabola.

Speaking of being without calculus, Pat Ballew explores finding derivatives without calculus.

John Cook at The Endeavour writes about how to differentiate a non-differentiable function.

Charles Wells considers the value of “naive proofs” — proofs that use “directly known facts” rather than traditional mathematics.

Danny Calegari at Geometry and the Imagination discusses a recent discovery regarding orthospectra of hyperbolic surfaces.

Qiaochu Yuan does some mindblowingly neat math on the bivariate generating function at his post Extracting the diagonal.

Rick Regan of Exploring Binary finds patterns in the last digits of the positive powers of two.

Edmund Harriss offers support to the thesis that “pure mathematics itself is a branch of concrete art.”

David Eppstein presents four open problems from a workshop on combinatorial geometry.

Maria H. Andersen has a new presentation based on her dissertation work entitled How can we measure Teaching and Learning in Math?

Brent Yorgey links together the hyperbinary sequence and the Calkin-Wilf tree, at the end of a 10 part series starting with this post. (It’s not as technical as it sounds!)

Mathspig has a 10-post series on the 10 Worst Mathematical Disasters. Start with #10 and work up from there; I had never heard about the one involving an Air Canada flight which ran out of fuel due to a unit conversion error.

If that’s not epic enough for you, Timothy Gowers has a (as of this writing) nine part series on the infamous P versus NP problem. This fascinating look into how a mathematician struggles with a problem starts here.

Dan MacKinnon at mathrecreation has found some interesting identities in Pascal’s Triangle.

The blog komplexify has a roundup of mathematical clocks. (Asking what’s wrong with clock #2 would make a good class opener.)

Matthew Kahle takes on topological Turán theory and the question: If a two-dimensional simplicial complex has vertices and faces, does it necessarily contain an embedded torus?

Dick Lipton lucidly explains a new paper on cheating the derivatives market.

Speaking of cheating, read a sordid tale of lies, money, and math games at the 2009 Philadelphia Sudoku Championship. (IMPORTANT NOTE: The case is still open. If you know anyone who was at the World Open Chess Tournament in 2006, please ask them if they can identify the person in the photo.)

Can an omnipotent deity create a rock he can’t lift? Terry Tao tackles mathematical equivalents when considering the “no self-defeating object” argument.

Colleen King teaches polynomials with the movement of a roller coaster at Learning in Mathland.

Nicolas Bourbaki was the pseudonym of a group of mathematicians that formed in 1935. In a four-part series, Lieven Le Bruyn investigates a cryptic wedding invitation found on the mathematician (and member of Bourbaki) André Weil which led to his arrest as a Russian spy.

And that’s a wrap! The next carnival to look forward to is Math Teachers at Play, which alternates Fridays with this carnival and is designed for “Tips, tidbits, games, and activities for students and teachers of preK-12 mathematics,” that is, mathematics appropriate for the younger set (as depicted by the young mathematician on the left).

Selections from the instructions for the 18th World Puzzle Championship

The 18th World Puzzle Championship starts today in Turkey and runs through this week.

Just to give a taste of how things will go, here are three selections from the full instructions for this year, available from the official website.

Circle Division

Draw given number of straight lines in the circle so that the sum of the numbers in all formed regions are the same. Lines should start and end on the circle perimeter and all formed regions should contain numbers.

Star Battle

Place exactly two stars in each row, each column and each outlined area. Stars have the size of one cell and cannot touch each other, not even diagonally.

Hang Up

Hang a rectangle (possibly a square) from its corner on each nail. All rectangles should have different perimeter lengths (not necessarily different areas) and their center of gravity should always be on the line going down from that nail. Rectangles cannot touch each other, not even diagonally.

As an extra bonus, here are the two Star Battle puzzles from Turkey’s own 2009 Puzzle Championship:

Cheating at the 2009 Philadelphia Inquirer Sudoku National Championship

This story was originally broken — and I mean broken, not just copied from elsewhere — by Thomas Snyder, 2-time winner of the World Sudoku Championship. The story is ongoing; perhaps you can help catch a cheat?

The 2009 Philadelphia Inquirer Sudoku National Championship occurred last week. It was done in three common rounds (with everyone solving puzzles in the same room), where the top 3 scores went on to a final round (depicted above).

The winners:

1. Tammy McLeod [in center of photo], Los Angeles ($10,000 and a seat on the U.S. team at the World Sudoku Championship in April in Philadelphia). She came in third last year. She finished the final round in 7 minutes, 41 seconds.

2. Thomas Snyder [in left of photo], Palo Alto, Calif. ($4,000). He finished in second place last year, and is a two-time world champion.

3. Eugene Varshavsky [in right of photo], Lawrenceville, N.J. ($3,000). He joined the competition during walk-up registration yesterday morning.

Thomas Snyder finished the final puzzle in 4:14, but made a transposition error allowing Tammy McLeod the win. The Inquirer has many more details.

The person on the right, Eugene Varshavsky, entered the competition in highly irregular circumstances. He skipped the first and second rounds altogether, arriving late. He then proceeded to finish the 3rd round in blazing time, qualifying him for the final. This is his grid at the end of the competition:

As Thomas Snyder writes:

It has 2 observable placements in it, both in row 5, and a suggestion that the 9 in R1C3 may be there too (eta: confirmed from other images now). It is however not the most focused image and does not tell how this grid got to this state, if erasing happened, etc. Still, having this for 8 minutes of work on the puzzle after demolishing 3 hard ones in 12-13 minutes to qualify is simply not possible.

Eugene wore the hood you see in the first picture the entire competition. There is only one picture of him with the hood down:

Also, according to the Philly Inquirer:

He gave his residence as Lawrenceville, N.J., but no one by that name is listed in the town, and efforts to discover his actual residence have turned up a trail of dead ends.

A LexisNexis search revealed that a Eugene Varshavsky in 2007 had given his residence as an address in Ewing, not far from Lawrenceville. But that address leads only to HB Machines, where proprietors said they knew of no such person.

This may be the second time this particular cheat may have shown up at a competition. Thomas Snyder again, regarding a “Varshavsky” at the World Open Chess Tournament:

In 2006, for example, a suspected incident of cheating occurred in the World Open Chess Tournament. Against Grandmaster Smirin, a relatively unknown player wearing a hat the whole time performed well beyond expectations and ranking to beat the Grandmaster. After some suspicion was raised, this unknown disappeared to a bathroom where after ten minutes he was searched and nothing was found. Under closer watch, without the possibility of using unallowed assistance, the performance of this player returned to more expected levels and he lost the following matches, coming nowhere near to the mastery he had demonstrated earlier.

Because of the mystery surrounding Varshavsky, this story is still developing. Was anyone who is reading this at the 2006 World Open? Does anyone recognize the man in the picture?

A reading experiment

In my post “When vocabulary isn’t the issue” I got the impression it was difficult to “step inside the head” of a student who misunderstood that particular problem from a reading perspective, so I thought I’d give an example that has a better chance of simulating the experience.

This is a puzzle called Slitherlink. I gave it to my students and asked them to attempt to work it out simply from the directions, but out of 100 or so students only a handful managed without extra assistance. (They were given that the word “adjacent” means “next to”, so the vocabulary was not a problem.)

I have given this to adults who also needed extra explanation, so don’t feel bad if you’re unsure at first what to do.

Draw a closed loop by connecting dots. Each number in the puzzle specifies how many adjacent sides are included in the loop. A zero means no part of the loop passes next to that number.

Here are four sample puzzles of the type:

This example is less than optimal in that (unlike the last post) I know how to teach reading for understanding here, but still, I’m curious: did you have difficulty, and how did you extricate yourself? How would you teach reading the instructions to this puzzle?

When vocabulary isn’t the issue

With English as a Second Language students it is well known that known how to do mathematics is only half the battle; the other half is understanding what problems mean in the first place.

It’s generally assumed that vocabulary is the central problem, and when the gaps are filled in the students will be fine.

[Released item from the 2006 AIMS Math test.]

Neither the mathematics nor the vocabulary are issues here. Yet I had students do poorly, because they could not understand what the question is asking.

How would you handle teaching (the reading of) this question? This is at the fringes of my ability; while I can get the students to figure out vocabulary from context or wade through academic language, I couldn’t find a good way to lead students to the answer here.

Carnival update

An excellent edition of Math Teachers at Play is up at mathrecreation.

The Carnival of Mathematics has now had a proper revival thanks to Mike Croucher; I will be hosting the next carnival here on November 6, 2009. You can use this submission form to send in your entries.

Everything math-related goes in here: proofs, explanations of basic concepts, puzzles, writings about math education, mathematical anecdotes, refutations of bad math, applications of math, reviews of popular math… Note that sufficiently mathematized portions of other disciplines, especially physics and computer science, are acceptable.

While Math Teachers at Play is dedicated to just K-12 appropriate mathematics, the Carnival of Mathematics is additionally open to research math of any level of depth.

Here are the prior carnivals I hosted:

Carnival of Mathematics #30
Carnival of Mathematics #43
Carnival of Mathematics #52
Math Teachers at Play #12

Commentary on excerpts from the Mathematics Core Common Standards

There is an intiative in the United States to form a common set of standards across the entire country; 48 states (excluding Texas and Alaska) have signed up.

Link to the Common Core Standards website

The plan is for these standards will eventually override any state standards being taught with right now.

The draft of the first phase of standards-making is up; comments are only being accepted from the public until October 20th. Hence there is something of an urgency in deciphering what’s going on.

In Language Arts, according to those in the know there’s enough worry to start sending a flood of concern.

I have read through the Mathematics standards and I’m not as worried, but there are some worthy excerpts to discuss.

The College and Career Readiness Standards for Mathematics will anchor the next phase of the Common Core State Standards Initiative: development of K–12 Mathematics Standards.

First note these are “general goal” standards, although there are clues (which I’ll get into in a moment) for what exact topics are of interest. Don’t expect to read the draft and get an exact curriculum yet.

The evidence tells us that in high performing countries like Singapore, the gap between what is taught and what is learned is relatively smaller than in Malaysia or the U.S. states. Malaysia’s standards are higher than Singapore’s, but their performance is much lower. One could interpret the narrower gap in Singapore as evidence that they actually use their standards to manage instruction; that is, Singapore’s standards were set within the reach of hard work for their system and their population. Singapore’s Ministry of Education flags its webpage with the motto, “Teach Less, Learn More.” We accepted the challenge of writing standards that could work that way for U.S. teachers and students: By providing focus and coherence, we could enable more learning to take place at all levels.

Those writing the standards appear to be updated on current research: there are far too many standards in the US.

An extra common point is textbooks in higher performings countries are the fraction of a size of the United States; however, I’m unsure how indicative this is of the actual material, since cutting pictures and paring down to the problems in US texts will also cut material significantly. However this point works, it suggests US textbooks may need some serious revision.

Overview of the Mathematical Content Standards

Number … Quantity … Expressions … Equations … Functions … Modeling … Shape … Coordinates … Probability … Statistics.

(I have cut the descriptions for the categories.)

An interesting approach: the categories in the document cut down to the objects of mathematics, rather than, say, grouping expressions and equations together as Algebra. One gets a strong sense the authors of the draft wanted to ensure the concepts were separate conceptually, even when manipulated in roughly the same way.

Part of the implication here is that each area will likely get some emphasis all the way down to K level, so to an extent even 3rd graders will have exposure to functions. (Likely as discussing input and output; I recall an educational game I played at elementary level where the “number machine” metaphor became quite clear to me.)

This also likely explains why “coordinates” are an entirely separate topic: the goal is to teach them at an early grade so we don’t have fumbling in later grades over if (0,5) is on the x-axis or the y-axis.

Extra-quirky: listing geometry as “shape”. Shape can be described by lines, and curves and so forth, but the lines can be studied in themselves without any shapes involved. I realize the intent here was to avoid the clumping word “geometry”, but I’m not sure any way around it. Here’s the full description:

Shape. From only a few axioms, the deductive method of Euclid generates a rich body of theorems about geometric objects, their attributes and relationships.

Technically I think they mean “geometric objects” as “shape”, but they’re trying to stick to single-word descriptions. While I understand the intent I worry teachers along the line may get confused, so a retitle of the category may be worthwhile.

Note also how the description immediately mentions Euclid. The draft gets a strong sense that curriculum needs a stronger proof basis, rather like how the NY Regents have swapped back to formal proofs very recently. This could cause issue with teachers who were never taught geometrical proofs at any level.

For example, systems of linear equations are covered by all states, yet students perform surprisingly poorly on this topic when assessed by ACT. We determined that systems of linear equations have high coherence value, mathematically; that this topic is included by all high performing nations; and that it has moderately high value to college faculty. Result: We included it in our standards.

One deep question of concern: if the authors are serious about cutting standards, which ones get cut?

This excerpt gives three hints to the overall strategy:

1. Weight is given to how highly colleges rate the skills in importance.
2. Topics are judged in comparison with high performing countries.
3. Standardized tests like the SAT and the ACT are additionally used as indicators.

Why were exponential functions selected for intensive focus in the Functions standard instead of, say, quadratic functions? What tipped the balance was the high coherence value of exponential functions in supporting modeling and their wide utility in work and life. Quadratic functions were also judged to be well supported by expectations defined under Expressions and Equations.

Two more hints:

4. Redundancies can be removed; quadratics are already covered in equations, so their coverage in functions is less important.
5. Topics that support applications are given higher priority.

Mathematically proficient students consider the available tools when solving a mathematical problem, whether pencil and paper, ruler, protractor, graphing calculator, spreadsheet, computer algebra system, statistical package, or dynamic geometry software. They are familiar enough with all of these tools to make sound decisions about when each might be helpful. They use mathematical understanding and estimation strategically, attending to levels of precision, to ensure appropriate levels of approximation and to detect possible errors. They are able to use these tools to explore and deepen their understanding of concepts.

(Emphasis mine.)

The writers of the standards draft seem to be of the technological bent, for this portion explicitly mentions the use of the technology. The practical ramification may be that when the common national test gets written, technology use will be included in a portion of the test.

How Mathematics Solved a Real Murder

At Mathspig blog:

A woman was found dead at the bottom of The Gap at Sydney’s Watson Bay in 1995. It wasn’t until 1997 that University of Sydney physicist, Rod Cross, was asked if the victim could have jumped off the cliff. In 1998, the coroner declared an open finding in the death of Byrne.

It wasn’t until 2003, however, that the police contacted Ross to check the maths. He said that she couldn’t have slipped or jumped. The case was reopened and in 2006 Byrne’s ex-boyfirend Wood was arrested in London and eventually found guilty of her murder. Wood was sentenced last year to 17 years in jail with a non-parole period of 13 years.

Why we are interested in this case, mathspigs, is because Cross, The Physicist, made the comment when asked during the trial that the maths involved was not rocket science but maths high school students would be able to master. Can we?

Solving the problem requires a relatively simple calculation with parabolas.

There is a book coming out this month based on the case, Evidence for Murder: How Physics Convicted a Killer by Rod Cross.

(Tip of the hat to simonjob.)

Is this a 6th grade geometry question?

The PSLE (Primary School Leaving Examination) occured in Signapore recently and it has some bloggers grumpy:

I swear, kids these days are learning stuff that are way more difficult than what we used to do back in primary school. Blame it on the big paper-chase that seems essential for surviving in this urban jungle we call Singapore, but really, by the time my kids come round to primary school, I don’t think I’ll be able to tutor them like my mother did to me.

Line segments AD, BD, and CD are congruent. What is angle ABC?

One blog claims a solution and another denies there is enough information for the solution to work.

Here’s the other question that has commenters sounding off:

Jim bought some chocolates and gave half of it to Ken. Ken bought some sweets and gave half of it to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. The ratio of Jim’s sweets to chocolates became 1:7 and the ratio of Ken’s sweets to chocolates became 1:4. How many sweets did Ken buy?

Simultaneous equations are required for a solution.

Also, one extra note from Wikipedia regarding the test:

The hard questions in the papers were to “filter” the average and below average students, as claimed by the Ministry of Education, only those who are truly the best can get good aggregate scores above 250.

What Can You Do With This (Giant Ants of Doom)

This is more of a How Can You Do This, because I bombed. Students were baffled. I’m not sure how to work the scaffolding on this one, even though I got the concept from a textbook. Any help appreciated. ADD: Suggestions to fix this lesson are in the comments.

Trailer for THEM!

Followed (after discussion) by:

Now, you may have some student give a clever answer using evolutionary biology, but at some point establish this is a Super-Science sort of situation, leading to:

Suppose you had a super-enlarger ray. How large could you make the ant so it still survives?

And here’s the textbook version (McDougal Littell, Algebra 2):

A common ant absorbs oxygen at a rate of about 6.2 milliliters per second per square centimeter of exoskeleton. It needs about 24 millilitres of oxygen per second per cubic centimeter of its body. An ant is basically cylindrical in shape, so its surface area S and volume V can be approximated by the formulas for the surface area and volume of a cylinder:

a. Approximate the surface area and volume of an ant that is 8 millimeters long and has a radius of 1.5 millimeters. Would this ant have a surface area large enough to meet its oxygen needs?

b. Consider a “giant” ant that is 8 meters long and has a radius of 1.5 meters. Would this ant have a surface area large enough to meet its oxygen needs?

So: no dice, even after simplifying the cylinder to a rectangular prism and exploring the 1×1x1 cube with the sides doubled first. (I could have led in with the Athenian Plague problem, which is roughly equivalent.)

In any case, students were paralyzed with indecision and I nearly had to do the problem for them to wrap it up. So, how could this work?