The troll and the dwarves

This Riddler puzzle is a classic! Can you save the dwarves from the troll?

A giant troll captures 10 dwarves and locks them up in his cave. That night, he tells them that in the morning he will decide their fate according to the following rules:

  1. The 10 dwarves will be lined up from shortest to tallest so each dwarf can see all the shorter dwarves in front of him, but cannot see the taller dwarves behind him.
  2. A white or black dot will be randomly put on top of each dwarf’s head so that no dwarf can see his own dot but they can all see the tops of the heads of all the shorter dwarves.
  3. Starting with the tallest, each dwarf will be asked the color of his dot.
  4. If the dwarf answers incorrectly, the troll will kill the dwarf.
  5. If the dwarf answers correctly, he will be magically, instantly transported to his home far away.
  6. Each dwarf present can hear the previous answers, but cannot hear whether a dwarf is killed or magically freed.

The dwarves have the night to plan how best to answer. What strategy should be used so the fewest dwarves die, and what is the maximum number of dwarves that can be saved with this strategy?

Extra credit: What if there are only five dwarves?

Here is my solution:
[Show Solution]

The honest prince

This Riddler puzzle is about randomly generating convex quadrilaterals.

You’re the most eligible bachelorette in the kingdom, and you’ve decided to marry a prince. The king has invited you to his castle so that you may choose from among his three sons. The eldest prince is honest and always tells the truth. The youngest prince is dishonest and always lies. The middle prince is mischievous and tells the truth sometimes and lies the rest of the time. Because you will be forever married to one of the princes, you want to marry the eldest (truth-teller) or the youngest (liar) because at least you know where you stand with each. But there’s a problem: You can’t tell the princes apart just by looking, and the king will grant you only one yes-or-no question that you may address to only one of the brothers.

What yes-or-no question can you ask that will ensure that you do not marry the middle prince?

Here is my solution:
[Show Solution]

Space race

This Riddler puzzle is about a game involving filling up the space on a square table using coins.

Two players are seated at a square table. The first player places a coin on the table, the second places a coin on the table, and they carry on placing coins one after another, with the only condition being that the coins are not allowed to touch. The winner is the person who places the final coin on the table, meaning that he or she fills the last remaining space between the other coins.

The table has to be larger than a single coin, and all the coins placed must be identically sized. If the players play optimally, is one of the two players guaranteed to win? If so, what is the winning strategy?

Need a hint?
[Show Solution]

Here is my solution:
[Show Solution]

Tangled wires

This Riddler puzzle is about tangled wires. Can you figure out how to untangle them?

There are N wires leading from the top of a bell tower down to the ground floor. But as wires tend to do, these have become hopelessly tangled. Good thing the wires on the top floor are all labeled, from 1 to N. The wires on the bottom, however, are not. (In retrospect, somebody probably should have anticipated a tangle or two.)

You need to figure out which ground-floor wire’s end corresponds to which top-floor wire’s end. (The bulk of the wiring is hidden behind a wall, so you can’t simply untangle them.) On the ground floor, you can tie two wire ends together, and you can tie as many of these pairs as you like. You can then walk to the top floor and use a circuit detector to check whether any two of the wires at the top of the tower are connected or not. What is the smallest number of trips to the top of the tower you need to take in order to correctly label all the wires on the bottom?

Here is my solution.
[Show Solution]

Adversarial map coloring

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]

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.
[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!

[Show Solution]