User talk:Bobthesmartypants/Problems


Problem 1: Bob is rolling a $6$-sided die. Every time he rolls a number that he has already rolled before, he rolls again. He stops when he has rolled all the numbers. What is the expected number of rolls it will take Bob?

Problem 2: Suppose you have a rectangular box, with side lengths $a$ and $b$, where $a$ and $b$ are positive integers. We launch a point-like ball from one of the vertices with an angular degree of $60^{\circ}$. The ball bounces off the sides of the box. Pretend there is no friction, drag, or anything else to slow down the ball. Prove or disprove that the ball won't ever hit a vertex again.

Problem 3: In a country, there is a particular way the cities inside are connected. One city has only one road leading out of it. One city has two roads leading out of it. Two cities have three roads leading out of it. Three cities have 5 roads leading out of it. In general, $F_n$ cities have $F_{n-1}$ roads leading out of it. Assume that all roads leading out of one city will enter another city, and vice versa. What values of $n$ are there such that this setup is possible?

Problem 4: What is the longest path possible on a truncated cube with edge lengths of $1$ if no edge can be used twice?

Problem 5: (Assume that there is no friction, and drag, and the ball follows the law of reflection.) Suppose we have an $n$-gon. There is a point-like ball at the midpoint of one of the sides of the $n$-gon (call that side Side A). It is launched to the midpoint of another side (call that Side B). Let $k$ be the number of sides clockwise to Side A but counterclockwise to Side B (not including Side A and Side B). Define $\delta_n(k)$ to be $k$ if the ball hits every side of the $n$-gon before returning to the launch point, and otherwise $0$. Find the closed form for the sum \[\sum^{n-2}_{k=0}{\delta_n(k)}\]

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