EXAMPLES of STUDENT WORK FROM THE FINAL

 

The final examination was a three hour examination and students worked independently. This differed dramatically from the conditions of the class and homework: Students collaborated in groups during class and had virtually as much time as they wanted to work on submitted problems. Despite the different environment and conditions nearly every student was able to submit one or more correct problems and every student had good ideas on several problems.

From here you can view the entire final examination and also a number of samples of student work. Each problem is accompanied with a short discussion of the approach they took to the problem and the completeness or rigor of their solution.

Product of a Pythagorean triple. In this problem parity and an indirect proof are used to show that for a Pythagorean triple (a,b,c) the product abc is even.

Product of differences. An application of the pigeonhole principle to a complex problem of divisbility.

Shortest paths. The symmetry principle is applied to find the shortest paths between pairs of points in the plane when the paths are required to touch one or more axes.

Isolated vertex and extended isolated vertex. In the first of these two solutions it is shown via the pigeonhole principle that either there is an isolated vertex or two vertices with the same degree. In the second, the student extends the problem and shows that even if there is an isolated vertex there are still two vertices with the same degree.

No solutions in integers. Unique factorization, solution of linear equations and parity are used to show that 30 cannot be expressed as the difference of two integral squares.

Covering a chessboard with dominos. A straightforward use of parity is used to shwo that a 7 x 7 chessboard cannot be covered with dominos (without overlapping).

Degree of vertex. A parity argument is used on the total degree of a graph to show that some vertex of a graph on nine vertices must have degree different from 3.

Equal checkers. Though not complete a novel representation of a problem to show that an equality occur.

A monochromatic rectangle. The pigeonhole principle is used to show that if the points of the plane are colored with 3 (100) colors there is still some rectangle whose vertices are all the same color.

Flippping coins. An invariance argument is used to show that if seven coins start all heads up and four can be flipped at a time then it is not possible to ever get all tails.

The number of odd numbers in a row of Pascal's triangle. This is an investigation type problem that calls for the student to do several examples and look for and uncover a pattern and then to prove their conjecture using mathematical induction.

A maximum problem from the Putnam competition. In this problem calculus is used to find the maximum of a function in this problem taken directly from the Putnam competition.

An area problem from the Putnam competition. In a simple equation is set up in order to solve an area problem taken directly from the Putnam competition.