Cranky Numbers: From the 3rd grade to Fermat's Last Theorem

What's the connection between 3rd grade math and Fermat's Last Theorem? My 3rd grader comes home with problems that ask him how many ways he can write a number, such as 4, prompting him to list such expressions as: 1+3, 2 x 2, and 8÷2. If you restrict the list to sums of non-negative integers, the list is short and finite: 0+4, 1+3, 2+2, 1+1+2, 1+1+1+1. The number of sums in the list is called the partition number; the fourth partition number is thus 5. A patient 3rd grader (if such existed) could find the partition number of any integer. Partition numbers are handy if you are a 3rd grade teacher making up problems to occupy your students or a particle physicist.

If you looked at a list of partition numbers (the first 20 are: 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490 627 more), you might notice that starting with the 4th partition number (5) every 5th number is divisible evenly by 5. Beginning with the 5th number (7), every 7th number divided evenly by 7. Perhaps not surprisingly, every 11th number after 11 is also divisible evenly by 11. The pattern ends there, but not the mystery. Indian mathematician Srinivasa Ramanujan recognized the patterns almost 100 years ago, but it took almost 40 years before Freeman Dyson explained them by inventing a function which he called the rank. Dyson's rank only explained the 5 and 7 patterns, roughly another 40 years would pass (is there a pattern here?) before the invention of the crank would account for the 11's.

As it turns out, there are more sequences buried in the list of partition numbers, you just have to know where to find them (a pattern based on those divisible by 13 begins with the 111,247th partition number). It also helps to have some of the techniques in number theory developed by Andrew Wiles to prove Fermat's Last Theorem. A proof that such patterns will exist for any prime number larger than 3 was published this year by Karl Mahlburg, a graduate student in math at the University of Wisconsin.

Extract DNA in your kitchen

When my kids think of DNA, they think of the classic picture of the double helix. But what does DNA really look like? A lot depends on how you look at it, and since most of us don't have a good scanning tunnelling microscope at home, we can't get pictures that look like this one from Lawrence Berkeley Labs.

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With a bit of patience, you can extract DNA in your kitchen, and see what the long polymer strands look like in the aggregate! The Genetic Science Learning Center at the University of Utah has developed a protocol for extracting DNA from split peas. Last summer my kids and several of their friends spent a morning in our kitchen pureeing peas and extracting the DNA. The white threads at the top of each test tube are the DNA. A great rainy day project - even if you don't have kids.

Building better ice cream and popcorn - who says physical chemistry is useless?

Physical chemistry, which I teach, has a certain reputation among chemistry majors: as difficult, dull, mathematically intensive, time consuming. There is even a bumper sticker that says "Honk if you passed p-chem!" When you are looking at the Maxwell relations in thermodynamics, it seems hard to imagine that p-chem has any impact on your daily life at all. But in reality, researchers at Purdue University are hot on the trail of better microwave popcorn and using physical chemistry to do it. [See Role of the Pericarp Cellulose Matrix as a Moisture Barrier in Microwaveable Popcorn; Agung S. Tandjung, Srinivas Janaswamy, Rengaswami Chandrasekaran, Adam Aboubacar, and Bruce R. Hamaker; Biomacromolecules].

Unpopped kernels in your popcorn are a pain, particularly when your kids pick them out and leave them in the living room! It turns out that unpopped kernels are even more of a problem in microwaved popcorn (is there any other kind anymore?). The key to getting popcorn to pop is is the structure of the outer hull (the pericarp), which is made of a biological polymer (which is why this was published in the journal Biomacromolecules). Pericarps in which the cellulose polymers exhibit a strongly crystalline structure pop better. The researchers used differential scanning calorimetry and x-ray crystallography to study the pericarp.

Prefer ice cream with your movie? Erich Windhab, at the ETH in Zurich (where Einstein once worked), used physical chemistry and physics to figure out how to make a smoother, richer ice cream - with fewer calories. You can now buy ice cream made with this process (Edy's Grand Light where I live). [See the article by Robert Kunzig in the June 2004 issue of Discover]