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!
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Randomized team drafting strategy

This Riddler Classic puzzle explores a randomized team drafting strategy designed to prevent teams from throwing games.

You are one of 30 team owners in a professional sports league. In the past, your league set the order for its annual draft using the teams’ records from the previous season — the team with the worst record got the first draft pick, the team with the second-worst record got the next pick, and so on. However, due to concerns about teams intentionally losing games to improve their picks, the league adopts a modified system. This year, each team tosses a coin. All the teams that call their coin toss correctly go into Group A, and the teams that lost the toss go into Group B. All the Group A teams pick before all the Group B teams; within each group, picks are ordered in the traditional way, from worst record to best. If your team would have picked 10th in the old system, what is your expected draft position under the new system?

Extra credit: Suppose each team is randomly assigned to one of T groups where all the teams in Group 1 pick, then all the teams in Group 2, and so on. (The coin-flipping scenario above is the case where T = 2.) What is the expected draft position of the team with the Nth-best record?

Here is my solution to the general case:
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While the expected draft position is not that difficult to characterize, one can also ask about the distribution of draft positions:
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Splitting a hundred dollar bill

This Riddler puzzle investigates a method for deciding who should get a $100 bill found on the ground. It leads to some interesting consequences…

You and four statistician colleagues find a \$100 bill on the floor of your department’s faculty lounge. None of you have change, so you agree to play a game of chance to divide the money probabilistically. The five of you sit around a table. The game is played in turns. Each turn, one of three things can happen, each with an equal probability: The bill can move one position to the left, one position to the right, or the game ends and the person with the bill in front of him or her wins the game. You have tenure and seniority, so the bill starts in front of you. What are the chances you win the money? What if there were N statisticians in the department?

Here is my solution to the first part, assuming five statisticians.
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Here is my solution to the second part, assuming $N$ statisticians.
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For the brave and curious, this next section explores connections between the problem and Fourier Transforms, complex analysis, and Chebyshev polynomials. Fair warning: advanced math!
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How many bananas can the camel carry?

This Riddler puzzle is a simple twist on a classic.

You have a camel and 3,000 bananas. You would like to sell your bananas at the bazaar 1,000 miles away. Your loyal camel can carry at most 1,000 bananas at a time. However, it has an insatiable appetite and quite the nose for bananas — if you have bananas with you, it will demand one banana per mile traveled. In the absence of bananas on his back, it will happily walk as far as needed to get more bananas, loyal beast that it is. What should you do to get the largest number of bananas to the bazaar? What is that number?

Here is my solution.
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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.
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Here, I go into more detail about bounding the optimal cake volume as the number of layers becomes large.
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Can you outrun the angry ram?

The Riddler puzzle this week appears simple at first glance, but I promise you it’s not!

You, a hard-driving sheep farmer, are tucked into the southeast corner of your square, fenced-in sheep paddock. There are two gates equidistant from you: one at the southwest corner and one at the northeast corner. An angry, recalcitrant ram enters the paddock from the southwest gate and charges directly at you at a constant speed. You run — obviously! — at a constant speed along the eastern fence toward the northeast gate in an attempt to escape. The ram keeps charging, always directly at you.

How much faster than you does the ram have to run so that he catches you just as you reach the gate?

Here is a very simple solution by Hector Pefo. Minimal calculus required!
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And here is my solution, which finds an equation for the path of the ram but requires knowledge of calculus and differential equations.
[Show Solution]

Pokémon Go Efficiency

This Riddler puzzle is about a topic near and dear to many hearts: Pokémon!

Your neighborhood park is full of Pokéstops — places where you can restock on Pokéballs to, yes, catch more Pokémon! You are at one of them right now and want to visit them all. The Pokéstops are located at points whose (x, y) coordinates are integers on a fixed coordinate system in the park.

For any given pair of Pokéstops in your park, it is possible to walk from one to the other along a path that always goes from one Pokéstop to another Pokéstop adjacent to it. (Two Pokéstops are considered adjacent if they are at points that are exactly 1 unit apart. For example, Pokéstops at (3, 4) and (4, 4) would be considered adjacent.)

You’re an ambitious and efficient Pokémon trainer, who is also a bit of a homebody: You wish to visit each Pokéstop and return to where you started, while traveling the shortest possible total distance. In this open park, it is possible to walk in a straight line from any point to any other point — you’re not confined to the coordinate system’s grid. It turns out that this is a really hard problem, so you seek an approximate solution.

If there are N Pokéstops in total, find the upper and lower bounds on the total length of the optimal walk. (Your objective is to find bounds whose ratio is as close to 1 as possible.)

Advanced extra credit: For solvers who prefer a numerical question with this theme, suppose that the Pokéstops are located at every point with coordinates (x, y), where x and y are relatively prime positive integers less than or equal to 1,000. Find upper and lower bounds for the length of the optimal walk, again seeking bounds whose ratio is as close to 1 as possible.

The problem of visiting a set of locations while minimizing total distance traveled is known as a Traveling Salesman Problem (TSP), and it is indeed a famous and notoriously difficult problem in computer science. That being said, bounding the solution to a particular TSP instance can be easy if we take advantage of its structure.

Here is my solution to the first part:
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Here is my solution to the second part:
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Should the grizzly eat the salmon?

This Riddler puzzle concerns a random process and sequential decision-making.

A grizzly bear stands in the shallows of a river during salmon spawning season. Precisely once every hour, a fish swims within its reach. The bear can either catch the fish and eat it, or let it swim past to safety. This grizzly is, as many grizzlies are, persnickety. It’ll only eat fish that are at least as big as every fish it ate before.

Each fish weighs some amount, randomly and uniformly distributed between 0 and 1 kilogram. (Each fish’s weight is independent of the others, and the skilled bear can tell how much each weighs just by looking at it.) The bear wants to maximize its intake of salmon, as measured in kilograms. Suppose the bear’s fishing expedition is two hours long. Under what circumstances should it eat the first fish within its reach? What if the expedition is three hours long?

Here is my solution:
[Show Solution]

The Traitorous Generals

This Riddler problem is a logic puzzle about liars and truth-tellers.

You are the emperor of Byzantium (lucky you!) and you have N generals working for you. You know that more than half of your generals are loyal, and the rest are traitors. You can ask any general about the loyalty of any other general: If the general you ask is loyal, he will answer correctly, but if he is a traitor he can answer however he likes. Your goal is to find one general you are absolutely certain is loyal while asking the fewest possible questions.

What is the minimum number of questions (in terms of N) that will guarantee a solution, and what strategy produces it?

There is a problem in cryptography known as the Byzantine Generals Problem, which has to do with achieving consensus in the presence of traitors that can sabotage communications. The Riddler problem above also involves liars and truth-tellers, but it’s a very different problem.

The following is adapted from a comment by Guy Moore. A similar solution that obtains the same final result was also found by Dmytro Taranovsky. Thank you both for your insights!

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