The First Day (1/5/00)

Before Class:

In just an hour I will begin my first class of Problem Solving. I intend to jump right in and after a brief introduction to get the students working on problems. I will give them a "syllabus", a survey, and a lengthy sheet of problems. More will be forthcoming. The first problems I will use in class today are the following:

1. 3 x 3 magic square.

2. Lockers problem

3. Getting a 1.5 meter sword on a plane if it can't be taken as hand luggage and no piece of luggage with a dimension greater than 1m will be accepted.

4. Number of ordered partitions problem.

5. 8 natural numbers < 16 show that there are three different pairs with a common difference.

6. Largest number of squares on an 8 x 8 chess board that can be colored green such that there is no totally green small "ell".

Will introduce some possible projects:

Buffon Needle Problem

The nonexistence of solutions to x^4 + y^4 = z^2.

Problem of the misplaced letters.

The number of triangulations of an n-gon

Expression of prime numbers of the form 4n+1 (uniquely) as a sum of squares.

Sum of the reciprocals of the squares of the natural numbers.

Probability that two natural numbers are relatively prime.

Classification of the regular solids

After Class

I have just returned from the first section. I appear to have about 14 students (one is auditing the course). I handed out the syllabus and the first problem set. They were enthusiastic. I showed some restraint with the magic square problem but ultimately gave some hints in the form of questions that might lead to systematic solution of the problem.

These questions were:

* Can you determine the common sum of the rows, columns, and diagonals?

* Knowing this common sum can you determine the value of the center square?

* Knowing the above information can you determine a solution?

* Can you show that the solution is essentially unique (up to reflections and rotations).

As a further hint I suggested students look at the problem of summing the first n natural numbers. (This is unnecessary since it is not difficult by hand to sum 1 + 2 + + 9 = 45).

Students also worked on the locker problem and the problem of attaching the houses to the power stations.

The groups worked adequately. Students come with the habit of working on their own and are not used to engaging in mathematical conversation with their peers. Thus it will take a while before there is really a synergy working in the groups. One student (Daniel who is also in linear algebra) had a nice discussion about finding sums of higher powers of consecutive integers and we discussed this briefly. I suggested he investigate this problem as an extended project after doing some reading in finite differences. He liked the idea since he has been interested in this question since eighth grade. All projects should work this well. Next class I will begin with students going to the board and discussing solutions to the magic square and locker problem. Then I have them work on ordered partitions and some pigeonhole problems.

Click here to go to problems for the first class.