# Past Math Club Discussion Problems

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## Fall 2007

### Thursday, September 6, 2007

How many different polynomios are there on an $n\times n$ grid? A polynomio is a connected figure of $2n-1$ squares (each connecting with at least one other square) that touches all 4 sides of the grid. (Problem asked by Donald Knuth in the Mathematical Monthly ~2001)

## Spring 2007

### Friday, February 2, 2007

What is the largest number that can be written in standard mathematical notation with 5 digits, letters or symbols?

Why is $\ln 2$ irrational?

## Fall 2006

### September 25, 2006

The "5 points on a sphere" problem (see below) has a counterexample with the phrase "exactly": consider 5 points equally spaced around the equator. (The original problem could be understood as saying "at least", in which case the solution is quite simple.)

2003 Putnam: If $a,b,c,A,B,C$ are constants with $a\ne 0\ne A$ and $|ax^2+bx+c|\le |Ax^2+Bx+C|$ for all real numbers $x$, then $|b^2-4ac|\le |B^2-4AC|$.

### September 18, 2006

3 Putnam problems: 1. A notch is cut out of a vertical circular log by two half-planes that meet with angle $\theta$ at a horizontal line through the center of the log. Show that the volume of the notch is smallest if the two half-planes make an angle of $\theta/2$ with the horizontal.

2. For $n>8$, $a=\sqrt{n}$, $b=\sqrt{n+1}$, decide which is bigger: $a^b$ or $b^a$?

3. For any 5 point on a sphere, show that there is a closed hemisphere with exactly 4 points on it.

### September 11, 2006

A geometric proof of the irrationality of e