Course webpage for Freshman Seminar 24i: Mathematical Problem Solving (Fall 2008)

If you find a mistake, omission, etc., please let me know by e-mail.

Your favorite problems or solutions
Here's a clearer picture of the background pattern...

September 15
Initial meeting:

September 22:

September 29:

Here are some more induction problems (and the LaTeX source).

October 6:

Here are some more generating function problems (and the LaTeX source).

October 13: [A University holiday, so no official class meeting. Instead Zach and I hold in effect an extended section, going over generating functions and the pending problems. Also, a more combinatorial proof of the formula for the sum of the squares of the entries of the n-th row of Pascal's triangle: once we know it's Binomial(2n,n), interpret Binomial(2n,n) as the number of NE paths from (0,0) to (n,n), and group them by which point at which the path crosses the diagonal (k,n-k).]

October 20:

October 27:
Solutions of problems postponed from previous weeks: Induction 2(ii) and 6; the last generating-function problem; the easy part of the Fermat point construction via complex numbers. Some simplifications: 2(ii) by telescoping; 6 is only barely an induction problem (need only the formula for 1+2+...+n); first part of 10(ii) is a special case of exponential convolution; for Fermat, write A-A'=A+rB+r2C where r=“cis(120)” is a complex cube root of unity, and likewise B-B' and C-C'.

Also: introduction to the 2-dimensional cross product, see
these further problems (and the LaTeX source). [corrected Nov.5]

November 3:

November 10:

Some problems on convexity and Jensen's inequality (and the LaTeX source). [corrected Nov.17]

November 17:

November 24:

December 1: Some considerations of the artificial problem-solving environment that is the coming Putnam exam; recent Putnam examples (full statement of problems and solutions here):

December 8:

December 15: