## Archive for November, 2007

### Chocolate-Peanut Butter Ice-cream Pie

November 30, 2007

In preparation for the dozen or so students who are coming over for dinner tonight in honor of Saturday’s Putnam exam, I made the pie pictured here. I made a bunch of pies, actually, because pie is really good. And then I thought “Pie is a mathy kind of dessert. We need a pie category on our Blog!” The premier recipe is this Chocolate-Peanut Butter Ice-Cream Pie by Heather Eckman. Click here for the recipe

### Writing Tests

November 29, 2007

An unexpected event in the last week has caused me to have to write three tests in less than two days. It often takes me at least that long to write one test, and I started thinking (now that they’re finished) about the test writing process. If you’ve ever written a test before, you probably know what I’m talking about, but if not, let me give you a little insight. More on the flip.

### Hashi (Bridges)

November 28, 2007

In the game Hashi or Hashiwokakero (Bridges/Chopsticks or Let’s Build Bridges!), you are given islands and you have to build bridges north-south or east-west between them. You are told the total number of bridges attached to each island, and you can build at most two bridges between any pair of islands. The bridges can’t cross, and it must be possible at the end to go from any island to any other island via some combination of bridges.

In other words, you are given the vertices of a graph and you have to construct vertical or horizontal edges. You are told the degree of each vertex, and you can have up to two edges between each pair of vertices. The final graph is planar and connected. Sound like fun? Then try it out here!

There are several mathematical questions you can ask:  What is the greatest number of bridges emanating from a single island?  What are the greatest/fewest number of bridges you could have for a given number or configuration of islands?  A quick search didn’t reveal nearly as much published on the mathematics of Hashi compared to, say, Sudoku, so this might be a fruitful area for some fun research!

The photograph of the bridge over the Struma River in Bulgaria is copyrighted © Nikola Gruev and is published on Wikipedia Commons under the terms of the GNU Free Documentation License.

November 27, 2007

I sensed (well, OK, I checked the blogstats) that the recent Math Magic posts on Mindreading were getting a lot of hits.   So for fun, here* is another mind-reading site.  In this one you are shown some cards, you pick one, and the computer uses amazing powers (mind-reading and/or listening skills) to determine which card you picked and remove it from the selection.  There’s no actual math in this one, although searching for it did reveal even more sites that do combine math and mind-reading, so there’s plenty of fodder for future entries on this topic!

* This site seems to have a lot of advertising on it, though I know I’ve seen less commercial sites with the same trick.

### How to tie your shoes, part 1

November 26, 2007

Reading Ξ’s post, I was struck at first by strong memories of my youth, learning various knots in scouting, and thought about how I don’t really tie knots very often anymore (Friday’s expedition to come home with a holiday tree notwithstanding).

But then again, most adults do tie knots on a regular basis. Many men wear cloth neck ornaments, and most men and women are faced with tying shoes — if not their own, then their childrens’.  [Continued]

### To Knot or Not to Knot

November 25, 2007

(I am so sorry about the title. I couldn’t help myself.)

Our local paper carried a short article this weekend about the International Guild of Knot Tyers, who just had an exhibition at the Museum of Shipping in Flensburg, Germany (the Flensburger Schiffahrtsmuseum). Upset that you missed it? Never fear! There’s still time to go to the Tenth Anniversary General Meeting and Convention of the North American Branch, November 29-December 2 in Florida. Their website contains a slew of photos from the past two meetings, and I confess I was rather jealous of all the cool knotted products around (not to mention the fancy designs that one could create from the string on the namebadges!)

### Math Magic: Mind-Reading in Person

November 24, 2007

Ask for a volunteer. The volunteer should pick a number; the number of digits doesn’t matter, but a 4-digit number tends to be big enough to be impressive without being overwhelming. The volunteer should write the number down, but not show you what it is. For example, the volunteer might secretly pick 2748. Click here to continue with the trick…

### Math Magic: Mind-Reading Web Sites

November 23, 2007

There are several sites online that claim to read your mind. A typical scenario is as follows:

You pick a two-digit number, add the digits, and subtract that sum from your original number. You then look up this difference in a chart of numbers [from 0 to 99] and make note of the associated symbol. The site will then tell you what symbol you noted.

You can play this game for yourself at (for example) MysticalBall.com or The Ghost Whisperer’s Crystal Ball on CBS’s website.

Magic on the Internet? A mathematical trick? See if you can figure it out (or post a comment if you need a hint!).

### Something to contemplate in the kitchen today

November 22, 2007

Consider the following advice for cooking the perfect Thanksgiving turkey:

The final temperature of the bird, after “resting” for 15 to 20 minutes, should be at least but not much more than 165 degrees Fahrenheit. The temperature will continue to rise 5 to 10 degrees after the turkey is removed from the oven. [theperfectturkey.com]

Does this mean that turkeys violate Newton’s Law of Heating and Cooling?

### And speaking of the Prisoner’s Dilemma…

November 22, 2007

My favorite example of the Prisoner’s Dilemma in literature occurs in the book Golem in the Gears by Piers Anthony (book 9 in the Xanth series).  In this book, a character examines the best strategy for playing the game multiple times, with different people (or beings, as the case may be).

### Tryptophan and Game Theory

November 21, 2007

You may have heard the fact that turkey contains tryptophan, an amino acid that the body uses in the production of serotonin. You may also have heard that it’s the tryptophan that causes post-Thanksgiving drowsiness.

This direct link is questionable (see, for example, Live Science). However, there’s a new connection afoot: tryptophan has been shown to affect trust and cooperation. And the study measuring that effect used Game Theory!

According to ABC News earlier today, neuroeconomists [did you even know there was such a job title?] used the Prisoner’s Dilemma to measure cooperation. Volunteers who drank a substance reducing their tryptophan levels were significantly less likely to cooperate than those with normal tryptophan levels, and they were also more suspicious of other players. Does this mean that eating turkey will make you more cooperative? Probably not (the levels are likely too low to make any difference), but we’re still happy to have an Actual Math Post related to Thanksgiving Day.

### Transforming Videos

November 20, 2007

Two amazing videos about mathematical transformations have been making their way across the Internet via YouTube and Google Video. Click to learn more and to see the videos!

### Nested Radicals Yield Cool Results

November 19, 2007

While browsing at Barnes & Noble, I came across Excursions in Number Theory by Ogilvy and Anderson. “Hey! I teach number theory,” I thought. So I flipped through the book for a while and came across the following question:

Let $n$ be an integer, and consider the expression

$\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}}$

Does this ever converge? If so, to what? The answer is somewhat surprising.

### Juxtapositions: The Rubik’s Hypercube

November 18, 2007

Many people are familiar with the Rubik’s Cube, the 3x3x3 cube with colored faces that can be moved out of position and then, ideally, twisted back into place. This toy, originally called the Magic Cube, was invented by Ernő Rubik in 1974 while he was a lecturer at the Academy of Applied Arts and Crafts in Budapest, Hungary. In 1980 the cube made its way to other countries and spawned, among other things, a one-season cartoon series Rubik, the Amazing Cube in which some kids use a magical come-to-life Rubik’s Cube to solve mysteries, while avoiding the mandatory evil magician who wants to steal the Cube.

### Juxtapositions: Fibonacci’s Triangle

November 17, 2007

What happens when you take two different ideas and put them together? That’s a driving question in mathematical research, and it can lead to some interesting and entertaining results.

One such juxtaposition is the concept of Fibonacci’s Triangle: a combination of the Fibonacci Sequence (1, 1, 2, 3, 5, 8, … where each number is the sum of the previous two) and Pascal’s Triangle*, shown to the left, which is created by placing “1”s along the outside and then filling the inside by adding two adjacent numbers and placing the sum between them on the next row.

In Fibonacci’s Triangle, according to Doug Ensley (see below), the string of “1”s along the outside is replaced by the Fibonacci sequence. The inside is filled in the same way as in Pascal’s triangle, with adjacent numbers in a row added together and the sum placed between them in the next row.

There are many questions that can be asked about Fibonacci’s triangle: Is there an explicit formula for the entries in each row the way there is for Pascal’s triangle? Does a Sierpinski-like pattern develop when you shade in the odd numbers the way it does for Pascal’s triangle? The answer to the first question is yes: see Doug Ensley’s article, “Fibonacci’s Triangle and Other Abominations” in the September 2003 issue of Math Horizons. For the second question, the best solution is to draw it out and see for yourself!

*You can see the picture in a 1665 copy of Pascal’s book Traité du triangle arithmétique here; other images are available from the Cambridge University Library Digital Image Collection.