Fourth Week of Instruction

(1/24, 1/26, 1/28)

Monday (1/24)I began the class taking stock of where we are, reviewing the methods and strategies we had talked about during the previous three weeks. I was asked how, when using the pigeonhole principle, to decide on what are the boxes (pigeonholes) to put the pigeons in. This led to a mini-lecture on equivalence relations and equivalence classes. Students broke into their assigned groups to choose a problem, from amongst the ones I suggested they work on together, and collaborated for the rest of the period. At first the groups were quiet as they individually read the problems and then became more animated as they argued for their preferred problem. Then relative quiet again as they began to think about the problem, followed by increasing conversation.

Wednesday (1/26)I handed out a new collection of problems but extended the period during which they could work on problems from the initial sheet with the caveat that problems which had been presented in class or solved by me could not be submitted unless it was an entirely new method. I then introduced a new technique," look for invariants", when something is changing look for how other things change and how and which things remain the same. I illustrated the method with a problem and then suggested a couple on the new collection, in particular, the cube problem and the polynomial problem and also point out that there may be some problems from the first sheet amendable to this line of argument, e.g. the 4 x 4 grid problem (Click here for examples of invariant problems). The stayed in these groups for the remainder of the period (and most went beyond class time) and there was lots of discussion, much more than the last time.

Friday (1/28)Today I explain another technique, the extremal principle. This can be a very powerful method and often leads to very short proofs. It can take many forms such as: choose the maximum element, choose the minimum element, or order the elements in ascending (or descending) order. It may also be the most difficult. I call students attention to the problems on the new sheet to which this method is applicable, for example the tennis tournament problem and a group or problems from the first sheet, for example, the water gun problem and infinite chess board problem (click here for extremal problems). They then break into groups and work on problems for the remainder of class.