Problems
A city planning committee contains a proportion \(p\) of members of the Even party, who never change their minds about anything, and a proportion \(1-p\) of members of the Odd party who change their minds completely at random (with probability \(r\)) between successive votes on the same issue. A randomly chosen committee member is noticed to have voted twice in succession in the same way. What is the probability that this member will vote in the same way next time?
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.
Show how to tile a \(4\times15\) rectangle with the twelve pentominoes.
Is it possible to tile a \(3\times20\) rectangle with the twelve pentominoes?
Show to how to cover the plane with this cube net:
Show how to tile the plane with this cube net:
