Inscribed triangles and tetrahedra

The following problems appeared in The Riddler. They involve randomly picking points on a circle or sphere and seeing if the resulting shape contains the center or not.

Problem 1: Choose three points on a circle at random and connect them to form a triangle. What is the probability that the center of the circle is contained in that triangle?

Problem 2: Choose four points at random (independently and uniformly distributed) on the surface of a sphere. What is the probability that the tetrahedron defined by those four points contains the center of the sphere?

Here is my solution to both problems:
[Show Solution]

Sticks in the woods

This Riddler puzzle is about making triangles out of sticks! Here is the problem:

Here are four questions about finding sticks in the woods, breaking them, and making shapes:

  1. If you break a stick in two places at random, forming three pieces, what is the probability of being able to form a triangle with the pieces?
  2. If you select three sticks, each of random length (between 0 and 1), what is the probability of being able to form a triangle with them?
  3. If you break a stick in two places at random, what is the probability of being able to form an acute triangle — where each angle is less than 90 degrees — with the pieces?
  4. If you select three sticks, each of random length (between 0 and 1), what is the probability of being able to form an acute triangle with the sticks?

For the tl;dr, here are the answers:
[Show Solution]

Here are detailed solutions to all four problems (with cool visuals!):
[Show Solution]

A tetrahedron puzzle

This post is about a 3D geometry Riddler puzzle involving spheres and tetrahedra! Here is the problem:

We want to create a new gift for fall, and we have a lot of spheres, of radius 1, left over from last year’s fidget sphere craze, and we’d like to sell them in sets of four. We also have a lot of extra tetrahedral packaging from last month’s Pyramid Fest. What’s the smallest tetrahedron into which we can pack four spheres?

Here is my solution:
[Show Solution]

Convex ranches

This Riddler puzzle is about randomly generating convex quadrilaterals.

Consider four square-shaped ranches, arranged in a two-by-two pattern, as if part of a larger checkerboard. One family lives on each ranch, and each family builds a small house independently at a random place within the property. Later, as the families in adjacent quadrants become acquainted, they construct straight-line paths between the houses that go across the boundaries between the ranches, four in total. These paths form a quadrilateral circuit path connecting all four houses. This circuit path is also the boundary of the area where the families’ children are allowed to roam.

What is the probability that the children are able to travel in a straight line from any allowed place to any other allowed place without leaving the boundaries? (In other words, what is the probability that the quadrilateral is convex?)

Here is my solution:
[Show Solution]

Cutting a circular table

This Riddler Classic puzzle is about cutting circles out of rectangles!

You’re on a DIY kick and want to build a circular dining table which can be split in half so leaves can be added when entertaining guests. As luck would have it, on your last trip to the lumber yard, you came across the most pristine piece of exotic wood that would be perfect for the circular table top. Trouble is, the piece is rectangular. You are happy to have the leaves fashioned from one of the slightly-less-than-perfect pieces underneath it, but there’s still the issue of the main circle. You devise a plan: cut two congruent semicircles from the perfect 4-by-8-foot piece and reassemble them to form the circular top of your table. What is the radius of the largest possible circular table you can make?

Here is my solution to the case of a general rectangular table. The result may surprise you!
[Show Solution]

Baking the optimal cake

This Riddler puzzle asks about finding the maximum-volume shape subject to constraints.

A mathematician who has a birthday coming up asks his students to make him a cake. He is very particular and asks his students to make him a three-layer cake that fits under a hollow glass cone he has on his desk. He requires that the cake fill up the maximum amount of volume under the cone as possible and that the layers of the cake have straight vertical sides. What are the thicknesses of the three layers of the cake in terms of the height of the glass cone? What percentage of the cone’s volume does the cake fill?

Here is my solution.
[Show Solution]

Here, I go into more detail about bounding the optimal cake volume as the number of layers becomes large.
[Show Solution]

Stop the alien invasion!

Today’s puzzle is from The Riddler, and has to do with spherical geometry.

A guardian constantly patrols a spherical planet, protecting it from alien invaders that threaten its very existence. One fateful day, the sirens blare: A pair of hostile aliens have landed at two random locations on the surface of the planet. Each has one piece of a weapon that, if combined with the other piece, will destroy the planet instantly. The two aliens race to meet each other at their midpoint on the surface to assemble the weapon. The guardian, who begins at another random location on the surface, detects the landing positions of both intruders. If she reaches them before they meet, she can stop the attack.

The aliens move at the same speed as one another. What is the probability that the guardian saves the planet if her linear speed is 20 times that of the aliens’?

Here is my solution for the case of interest, where the guardian is faster than the intruders.
[Show Solution]

Here is a partial solution for the more complicated case where the guardian is slower than the intruders.
[Show Solution]

Cutting polygons in half

This Riddler puzzle is about cutting polygons in half. Here is the problem:

The archvillain Laser Larry threatens to imminently zap Riddler Headquarters (which, seen from above, is shaped like a regular pentagon with no courtyard or other funny business). He plans to do it with a high-powered, vertical planar ray that will slice the building exactly in half by area, as seen from above. The building is quickly evacuated, but not before in-house mathematicians move the most sensitive riddling equipment out of the places in the building that have an extra high risk of getting zapped.

Where are those places, and how much riskier are they than the safest spots? (It’s fine to describe those places qualitatively.)

Extra credit: Get quantitative! Seen from above, how many high-risk points are there? If there are infinitely many, what is their total area?

Here is my solution:
[Show Solution]

And here is a bonus interactive graphic showing the solution

The puzzle of the picky eater

Today’s Riddler post is a neat problem about calculating areas.

Every morning, before heading to work, you make a sandwich for lunch using perfectly square bread. But you hate the crust. You hate the crust so much that you’ll only eat the portion of the sandwich that is closer to its center than to its edges so that you don’t run the risk of accidentally biting down on that charred, stiff perimeter. How much of the sandwich will you eat?

Extra credit: What if the bread were another shape — triangular, hexagonal, octagonal, etc.? What’s the most efficient bread shape for a crust-hater like you?

Here is my solution:
[Show Solution]

Overflowing martini glass

This Riddler puzzle is all about conic sections.

You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the drink reaches some fraction p of the way up its side. When tipped down on one side, just to the point of overflowing, how far does the drink reach up the opposite side?

Here is my solution:
[Show Solution]