Problems

Suppose that $p$ is a prime number. How many numbers are there less than $p^2$ that are relatively prime to $p^2$?

How many cuboids are contained in an $n\times n\times n$ cube? For example, we’ve got $n^3$ cuboids of size $1\times 1\times 1$, and obviously just $1$ of size $n\times n\times n$ (which is the whole cube itself). But we also have to count how many there of size $1\times 1\times 2$, $1\times 2\times 3$, and several more.

In the $6\times 7$ large rectangle shown below, how many rectangles are there in total formed by grid lines?

Simplify $F_0-F_1+F_2-F_3+...-F_{2n-1}+F_{2n}$, where $n$ is a positive integer.

Explain why a position $g$ is a winning position if there is a move that turns $g$ into a losing position. On the other hand, explain why a position is a losing position if all moves turns it into a winning position.

A technique that can be used to completely solve certain games is drawing game graphs. Given a game $G$, we draw an arrow pointing from a position $g$ to a position $h$ if there is a move from $g$ to $h$.

As a simple example, the game graph of $\text{Nim}(2)$ is shown below.

Draw the game graph of $\text{Nim}(2,2)$. Is $\text{Nim}(2,2)$ a winning position or losing position?

Is $\text{Nim}(2,5)$ a winning position or a losing position?

Let $x,y$ be nonnegative integers. Determine when $\text{Nim}(x,y)$ is a losing position and when it is a winning position.

Is $\text{Nim}(1,2,3)$ a winning position or a losing position?

Is $\text{Nim}(1,2,4,5,5)$ a winning position or a losing position?