Problems

A bin has \(2\) white balls and \(3\) black balls. You play a game as follows: you draw balls one at a time without replacement. Every time you draw a white ball , you win a dollar, but every time you draw a black ball, you lose a dollar. You can stop the game at any time. Devise a strategy for playing this game which results in an expected profit.

Take a (finite) set \(S\), say \([n]\) and a random function \(f:S\to S\). What’s the distribution of the limiting size of the image of the iterates of \(f\)?

That is, \(\lim_{N\to\infty}|f^N([n])|\)

By random, let \(i\in[n]\). Each \(f(i)\) is independently and identically distributed as uniform random variables on \([n]\). One can also think of it as \(f\) is taken uniformly from the \(n^n\) possible functions \([n]\to[n]\).

Today we’ll look at 3-dimensional shapes, including their volumes and surfaces areas. One special kind are the Platonic Solids - the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

In the picture below, there are the \(12\) pentominoes. Is it possible to tile a \(6\times10\) rectangle with them?

Show how to tile a \(5\times12\) rectangle with the twelve pentominoes.