## The slow car chase

This week’s Fiddler is a problem about probability and a slow car chase!

You and your pursuer are stopped at traffic lights one block away from each other, as shown below. For both cars, it takes 1 minute to get to each subsequent light, and there is a 50-50 chance any light will be red upon arrival (independent of what happened previously). If a light is red, you must wait one minute before it turns green. What is the probability you will make it to the city limits without being caught by your pursuer?

Extra Credit
In the same scenario as before, imagine there are infinitely many lights. On average, how many minutes will it be until you are ultimately caught?

My solution:
[Show Solution]

## Pinball probability

This week’s Fiddler is a challenging probability question.

You’re playing a game of pinball that includes four lanes, each of which is initially unlit. Every time you flip the pinball, it passes through exactly one of the four lanes (chosen at random) and toggles that lane’s state. So if that lane is unlit, it becomes lit after the ball passes through. But if the lane is lit, it becomes unlit after the ball passes through. On average, how many times will you have to flip the pinball until all four lanes are lit?

Extra credit: Instead of four lanes, now suppose your pinball game has $n$ lanes. And let’s say that $E(n)$ represents the average number of pinball flips it takes until all $n$ lanes are lit up. Now, each time you increase the number of lanes by one, you find that it takes you approximately twice as long to light up all the lanes. In other words, $E(n+1)$ seems to be about double $E(n)$. But upon closer examination, you find that it’s not quite double. Moreover, there’s a particular value of $n$ where the ratio $E(n+1)/E(n)$ is at a minimum. What is this value of $n$?

My solution:
[Show Solution]

## The weaving loom problem

This week’s Fiddler is a classic problem.

A weaving loom consists of equally spaced hooks along the x and y axes. A string connects the farthest hook on the x-axis to the nearest hook on the y-axis, and continues back and forth between the axes, always taking up the next available hook. This leads to a picture that looks like this:

As the number of hooks goes to infinity, what does the shape trace out?

Extra credit: If four looms are rotated and superimposed as shown below, what is the area of the white region in the middle?

My solution:
[Show Solution]

## Betting on football with future knowledge

This week’s Fiddler is a football-themed puzzle with a twist: you can see the future! Sort of.

My solution:
[Show Solution]

## Möbius prism

This week’s Fiddler is a problem about a multi-sided Möbius prism:

Consider an three-dimensional prism whose bases are regular N-gons. I twist it and stretch it into a loop, before finally connecting the two bases. Suppose that my twist is by a random angle, such that the two bases are aligned when they are coonected. Among all whole number values of N less than or equal to 1,000, for which value of N will a randomly twisted regular N-gon prism have the most distinct faces, on average?

My solution:
[Show Solution]