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]

Ellipse packing

You’ve heard of circle packing… Well this week’s Riddler Classic is about ellipse packing!

This week, try packing three ellipses with a major axis of length 2 and a minor axis of length 1 into a larger ellipse with the same aspect ratio. What is the smallest such larger ellipse you can find? Specifically, how long is its major axis?

Extra credit: Instead of three smaller ellipses, what about other numbers of ellipses?

My solution:
[Show Solution]

Randomly cutting a sandwich

This week’s Riddler Classic is geometry puzzle about randomly slicing a square sandwich.

I have made a square sandwich, and now it’s time to slice it. But rather than making a standard horizontal or diagonal cut, I instead pick two random points along the perimeter of the sandwich and make a straight cut from one point to the other. (These points can be on the same side.)

What is the probability that the smaller resulting piece has an area that is at least one-quarter of the whole area?

My solution:
[Show Solution]

Perfect pizza sharing

This week’s Riddler Classic is about how to cut a pizza to achieve precise area ratios between the slices.

Dean made a pizza to share with his three friends. Among the four of them, they each wanted a different amount of pizza. In particular, the ratio of their appetites was 1:2:3:4. Therefore, Dean wants to make two complete, straight cuts (i.e., chords) across the pizza, resulting in four pieces whose areas have a 1:2:3:4 ratio.

Where should Dean make the two slices?

Extra credit: Suppose Dean splits the pizza with more friends. If six people are sharing the pizza and Dean cuts along three chords that intersect at a single point, how close to a 1:2:3:4:5:6 ratio among the areas can he achieve? What if there are eight people sharing the pizza?

My solution:
[Show Solution]

To jump straight to the results:
[Show Solution]

How high should you climb up the tower?

This week’s Riddler classic is a neat geometry problem.

Two people climb two of the tallest towers on an planet, which happen to be in neighboring cities. You both travel 100 meters up each tower on a clear day. Due to the curvature of the planet, they can barely make each other out. The first person returns to the ground floor of their tower. How high up their tower must the second person be you can barely make each other out again?

My solution:
[Show Solution]

Desert escape

This week’s Riddler classic is about geometry and probability, and desert escape! Here is the (paraphrased) problem:

There are $n$ travelers who are trapped on a thin and narrow oasis. They each independently pick a random location in the oasis from which to start and a random direction in which to travel. What is the probability that none of their paths will intersect, in terms of $n$?

My solution:
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Cone crawling

This week’s Riddler Classic is a geometry problem about traversing the surface of a cone

The circular base of the cone has a radius of 2 meters and a slant height of 4 meters. We start on the base, a distance of 1 meter away from the center. The goal is to reach the point half-way up the cone, 90 degrees around the cone’s central axis from the start, as shown. What is the shortest path?

Here is my solution:
[Show Solution]

Visualize the vertex

This week’s Riddler Classic is a neat geometry problem about

Suppose you have two distinct points anywhere on the coordinate plane. If I tell you that a parabola with a vertical line of symmetry passes through those two points, where on the plane could that parabola’s vertex be?

Here is my solution:
[Show Solution]

Polarization Puzzle

This week’s Riddler Classic is about light polarization.


When light passes through a polarizer, only the light whose polarization aligns with the polarizer passes through. When they aren’t perfectly aligned, only the component of the light that’s in the direction of the polarizer passes through. For example, here is what happens if you use two polarizers, the first at 45 degrees, and the second at 90 degrees. The length of the original vector is decreased by a factor of 1/2.

I have tons of polarizers, and each one also reflects 1 percent of any light that hits it — no matter its polarization or orientation — while polarizing the remaining 99 percent of the light. I’m interested in horizontally polarizing as much of the incoming light as possible. How many polarizers should I use?

Here is my solution:
[Show Solution]

Inscribed hexagons

This week’s Riddler Classic is a geometry problem involving inscribed hexagons.

The larger regular hexagon in the diagram below has a side length of 1. What is the side length of the smaller regular hexagon?
If you look very closely, there are two more, even smaller hexagons on top. What are their side lengths?

Here is my solution:
[Show Solution]