Why is there no Gömböc in two dimensions?

Here's an argument to show why there is no convex and homogeneous two-dimensional shape with one stable and one unstable point of equilibrium.

An exhaustive algorithm for finding mixed tilings

\textbf{Step 1} Let $k$ be either 3, 4, 5, or 6. \textbf{Step 2} Set $c:=\frac{k-2}{2}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}.$ \textbf{Step 3} Set $n$ to be the biggest whole number less than or equal to $\frac{c}{k}$. Note that not all $n_i$ can be bigger than $n$, for otherwise $$\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}<c.$$ \textbf{Step 4} Suppose $n_1

We are proud to present the winners of the Plus New Writers Award 2009.

Would you prefer a game with a 90% chance of winning, or one with a 10% chance of losing?

The movie below shows the time evolution of a solution of the Cahn-Hilliard equation. Starting from an almost homogenous mixture of two alloys, the equation dynamics separate the mixture into its pure components (black and white). Later in time small regions containing the same alloy melt together and build larger and larger regions until the equation reaches a stationary state.

Step 3

Show that if N does not lie in our array, then 2N+1 is prime.

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