## Archipelago

A graph theory problem from the Riddler blog. Here it goes:

You live on the volcanic archipelago of Riddleria. Your archipelago is connected via a network of bridges, forming one unified community. In an effort to conserve resources, the ancient Riddlerians who built this network opted not to build bridges between any two islands that were already connected to the community otherwise. Hence, there is exactly one path from any one island to any other island.

Each island contains exactly one volcano. You know that if a volcano erupts, the subterranean pressure change will be so great that the volcano will collapse in on itself, causing its island — and any connected bridges — to crumble into the ocean. Remarkably, other islands will be spared unless their own volcanoes erupt. But if enough bridges go down, your once-unified archipelagic community could split into several smaller, disjointed communities.

If there were N islands in the archipelago originally and each volcano erupts independently with probability p, how many disjointed communities can you expect to find when you return? What value of p maximizes this number?

Here is my solution:
[Show Solution]

## Pool hall robots

This Riddler puzzle is about arranging pool balls using a robot!

You own a start-up, RoboRackers™, that makes robots that can rack pool balls. To operate the robot, you give it a template, such as the one shown below. (The template only recognizes the differences among stripes, solids and the eight ball. None of the other numbers matters.)

First, the robot randomly corrals all of the balls into the wooden triangle. From there, the robot can either swap the location of two balls or rotate the entire rack 120 degrees in either direction. The robot continues performing these operations until the balls’ formation matches the template, and it always uses the fewest number of operations possible to do so.

Using the template given above — a correct rack for a standard game of eight-ball — what is the maximum number of operations the robot would perform? What starting position would yield this? How about the average number of operations?

Extra credit: What is the maximum number of operations the robot would perform using any template? Which template and starting position would yield this?

Here is my solution:
[Show Solution]

Consider four towns arranged to form the corners of a square, where each side is 10 miles long. You own a road-building company. The state has offered you \$28 million to construct a road system linking all four towns in some way, and it costs you \$1 million to build one mile of road. Can you turn a profit if you take the job?

Extra credit: How does your business calculus change if there were five towns arranged as a pentagon? Six as a hexagon? Etc.?

Here is a longer explanation:
[Show Solution]

Here is the solution with minimal explanation:
[Show Solution]

This Riddler problem considers the classical map-coloring problem with an adversarial twist! One player draws countries and the other player colors them.

Allison and Bob decide to play a map-coloring game. Each turn, Allison draws a simple closed curve on a piece of paper, and Bob must then color the interior of the “country” that curve creates with one of his many crayons. If the new country borders any pre-existing countries, Bob must color the new country with a color that is different from the ones he used for the bordering ones.

Allison wins the game when she forces Bob to use a sixth color. If they both play optimally, how many countries will Allison have to draw to win?

Here is my solution:
[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:
[Show Solution]

Here is my solution to the second part:
[Show Solution]