Down With Geometry

One of my favorite bloggers, Kevin Drum, decided to relieve the tedium of our current political anarchy by whacking the hornets’ nest of the high school mathematics curriculum, in particular the subject of plane geometry.  You can tell from the tag list on my blog that I hold plane geometry in high regard and can’t let this gibe pass without some rebuttal, futile as it may be.  Actually, I am not going to weigh in on the general issue of the current math curriculum that much, but rather make a few observations from my own experience over the years as it relates to Kevin’s post

Curve Making Puzzle

Here is a problem from Five Hundred Mathematical Challenges that I indeed found quite challenging.

“Problem 235. Two fixed points A and B and a moving point M are taken on the circumference of a circle. On the extension of the line segment AM a point N is taken, outside the circle, so that lengths MN = MB. Find the locus of N.

Since one of the first hurdles I faced with this problem was trying to figure out what type of shape was being generated, I thought I would omit my usual drawings illustrating the problem statement.  There turned out to be a lot of cases to consider, but the result was most satisfying.  I also included the case when N is inside the circle.  Again Visio was my main tool to handle all the examples with the concomitant requirement to prove whatever Visio suggested.

See the Curve Making Puzzle

The Thoth Maneuver

This is a nice puzzle from Clifford Pickover in the 1996 Discover magazine’s Brain Bogglers.

“Thoth, ancient Egyptian god of wisdom and learning, has abducted Ahmes, a famous Egyptian scribe, in order to assess his intellectual prowess. Thoth places Ahmes before a large funnel set in the ground. It has a circular open­ing 1,000 feet in diameter, and its walls are quite slippery. If Ahmes attempts to enter the funnel, he will slip down the wall. At the bottom of the funnel is a sleep-inducing liquid that will instantly put Ahmes to sleep for eight hours if he touches it.

As shown in the illustration, there are two ankh-shaped towers. One stands on a cylindrical platform in the center of the fun­nel. The platform’s surface is at ground level. The distance from the platform’s surface to the liquid is 500 feet. The other ankh tower is on land, at the edge of the funnel.

Thoth hands Ahmes two objects: a rope 1,006.28 feet in length and the skull of a chicken. Thoth says to Ahmes, ‘If you are able to get to the central tower and touch it, we will live in harmony for the next millennium. If not, I will detain you for fur­ther testing. Please note that with each passing hour, I will decrease the rope’s length by a foot.’

How can Ahmes reach the central ankh tower and touch it? ”

See the Thoth Maneuver

The Percentage Trap

Twitter comments to the recently released GDP growth numbers for the Third Quarter reminded me of an old trap regarding percentages.  The financial and technical articles were accurate, but the comments by Twitter users often reflected the pitfall.

After a Second Quarter annualized US GDP fall of about 33%, the Third Quarter showed a gain of about 33%.  So some commenters thought the gain canceled the previous loss.

Rufus Puzzle

Again we have a puzzle from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).  This one is quite a bit more challenging, at least for me.

“When Holmes and I met with Wiggins one afternoon, he was accompanied by a rather scrappy-looking mutt, who eyed me with evident suspicion.

‘This is Rufus,’ Wiggins said. ‘He’s a friend.’

‘Charmed,’ I said.

‘He’s very energetic,’ Wiggins told us. ‘Just this morning, he and I set out for a little walk.’

At the word ‘walk’, the dog barked happily.

‘When we set out, he immediately dashed off to the end of the road, then turned round and bounded back to me. He did this four times in total, in fact. After that, he settled down to match my speed, and we walked the remaining 81 feet to the end of the road at my pace. But it seems to me that if I tell you the distance from where we started to the end of the road, which is 625 feet, and that I was walking at four miles an hour, you ought to be able to work out how fast Rufus goes when he’s running.’

‘Indeed we should,’ said Holmes, and turned to look at me expectantly.

What’s the dog’s running speed?”

See the Rufus Puzzle

Circle-Halving Zigzag Problem

This is another delightful Brainteaser from the Quantum math magazine.

“All the vertices of a polygonal line ABCDE lie on a circumference (see the figure), and the angles at the vertices B, C, and D are each 45°.

Prove that the area of the blue part of the circle is equal to the area of the yellow part. (V. Proizvolov)”

I especially liked this problem since I was able to find a solution different from the one given by Quantum.  Who knows how many other variations there might be.

Swallowing Elephants

This is a simple logic puzzle from one of Ian Stewart’s many math collections.

1. Elephants always wear pink trousers.
2. Every creature that eats honey can play the bagpipes.
3. Anything that is easy to swallow eats honey.
4. No creature that wears pink trousers can play the bagpipes.

Therefore:

Elephants are easy to swallow.

Is the deduction correct, or not?

Pool Paving Problem

In my search for problems I decided to purchase Dan Griller’s GCSE problem book  mentioned in the Cube Roots Problem.  I am still a bit confused about the purpose of the GCSE exam and who it is for, since the other problems in Griller’s book are often as challenging or more so than the cube roots problem.  It is hard to believe students not pursuing college level degrees could solve these problems.  (Grades 8 and 9 referred to in the subtitle of the book must indicate something other than US grades 8 and 9, since the exams are aimed at 16 year-olds, not 13 and 14 year-olds.)

Supposedly the problems in Griller’s book are nominally arranged in increasing order of difficulty from problem 1 to problem 75.  However it seemed to me that there were challenging problems scattered throughout and the last problem was not all that much harder than earlier ones.  And many of them had a whiff of Coffin Problems—they seemed impossible at first (Problem 44: Construct a 67.5° angle!).  I don’t know how many problems are on the exam or how long the exam is, but anyone taking a timed exam does not have the leisure to mull over a problem.  The student only has a few minutes to come up with an approach and clever insights are rare under the circumstances.  Anyway, here is the last problem in the book.

“Problem 75.  A square pond of side length 2 metres is to be surrounded by twelve square paving stones of side length 1 metre.

(a)  The first design is constructed with a circle whose centre coincides with the centre of the pond.  Calculate exactly the total dark grey area for this design.

(b)  The second design is similar.  Calculate exactly the total dark grey area for this second design.”

See the Pool Paving Problem