# Puzzles

This page features a collection of some of my favourite puzzles. If you have a new one in a similar vein, please drop me a note. I apologize to those of you who have written requesting solutions or hints, but I do not have the time to respond.

The burning strings problem: Thanks to Gilad Yarnitzky of Petach Tikva, Israel
You have two strings, each of which burns in exactly one hour, although not at a constant rate. In other words, it may take more or less than half an hour to burn one half of either string. How can you use these strings to measure three quarters of an hour?

The ball-drop problem: Thanks to Gilad Yarnitzky of Petach Tikva, Israel
You are given two identical glass balls and asked to find the highest floor of a 100-story building from which it is safe to drop these balls. In the process, you are permitted to break both balls. In the worst case, what is the lowest number of times you have to drop a ball from the building?

The handshake problem: Thanks to Gilad Yarnitzky of Petach Tikva, Israel
Five non-same-sex couples, including the host and his wife, attend a party. Throughout the evening, as introductions are made, various partygoers shake hands. However, couples that are together do not shake hands with each other. At the end of the evening, the host asks the nine other attendees how many hands they shook and obtains nine different answers. How many hands did the host shake and how many hands did his wife shake?

The safe transport problem: Thanks to Erez Strauss of Tel Aviv, Israel
Two kings wish to exchange gifts via an untrustworthy messenger, who has a tendency to steal whatever is sent in an unlocked chest. Thus, the kings decide to padlock the chest. How can each king allow the other to open the chest, without the messenger stealing the contents?

The camel division problem: Thanks to Gilad Yarnitzky of Petach Tikva, Israel
A recently deceased bedouin's family gathers to divide the inheritance. His will leaves 17 camels to be divided 1/2 to the first son, 1/3 to the second son, and 1/9 to the third son. What do they do?

The tree-planting problem: Thanks to Daniel Sud of McGill
Ten trees are planted to form five rows of four trees each. How are they arranged?

The two boxes problem: Thanks to John Thomas of NYNEX
You have two locked boxes, each of which contains the unique key for the other box, and openable only via its key lock. You can open both boxes... how?

Prime math: The sum of two primes is 999. What is their product?

The cheating husbands problem: Thanks to Michael Stumm of the University of Toronto
A certain village contains a number of married couples, of which k husbands are cheating on their wives. Every woman is aware of all the cheating taking place, expect for the infidelities of her own husband. In order to uphold a strict morality, the women of the village make a pact: any woman who learns that her husband has been cheating will kill him that night and dump his body in the town square for all to see. However, because no one wants to tell another woman that her husband is being unfaithful, this information is never communicated, and so, the cheating continues. Some time later, at a town meeting, the chief announces, "I want the cheating in this village to stop." Then, k-1 nights pass uneventfully, but on the kth night, all k cheating husband are killed by their wives. How did this happen?

The pasta loops problem: Thanks to Carlos Cordero of Palo Alto, CA, and credit to a Stanford University qualification exam
You are given a potful of n spaghetti noodles and charged with the task of sticking your hands in, randomly pulling up two ends (these may be two ends of the same noodle), tying them together, and dropping them back in, until all the ends have been tied. How many loops do you expect to produce?

The wandering hiker problem: The classic puzzle is worded as follows: You go for a hike, walking one mile south, one mile west, and one mile north, and end up back at your starting point. Where are you? For those who already know the answer, the follow-up question is where else could you be? Can you generalize the answer?

The egg basket problem: Thanks to Ghassan Dally of Toronto, ON
I have a basket of eggs and give half of these plus half an egg to person A, then half of the remaining plus half an egg to person B, then half of the remaining plus half an egg to person C. My basket is now empty and no eggs were cracked open during the process. How many eggs were in the basket initially?

The island is on fire problem: Thanks to Jeff Mooallem of Toronto, ON
You find yourself in the middle of a forest-covered island, surrounded by shark infested waters. A fire has started on the eastern side, and a steady westerly wind is spreading the fire towards your position. How do you survive?

The Monty Hall problem: You have three closed doors, each leading to a separate room. One of the rooms contains a large prize, while the other two are empty. You are asked to stand in front of any door and offered whatever is in the corresponding room. However, before the contents of that room are revealed to you, the gamemaster will open up one of the remaining two doors to expose an empty room (the gamemaster has full knowledge of the contents of each room and will not show you the winning room). After the empty room has been exposed, you are then given the option of sticking with your original choice, or selecting another door. What do you do? Hint: The intuitive answer ("it doesn't matter") is wrong. But why?

The modified Monty Hall problem: Thanks to Carlos Cordero of Palo Alto, CA
Back to our original puzzle, the Monty Hall problem, assume that each of the doors is labelled with the probability of the prize being found behind it, the numbers being 9/20, 4/20, and 7/20. As before, you are asked to stand in front of any door and offered whatever is in the corresponding room. I will then open one of the remaining two doors to expose an empty room, and offer you the choice of switching doors. What strategy should you take to maximize your profit potential?

The two condoms problem: Thanks to Andrea Freeman of Vancouver, BC
A man would like to have safe sex with three women, any of whom may be carrying an STD. Given two condoms, how can he do so, while ensuring that no STD is passed from one woman (or possibly himself) to another (or to himself)?

Variant on the two condoms problem: Thanks to Erez Strauss of Tel Aviv, Israel
Two men and two women would like to have safe sex in all four hetrosexual permutations. Given two condoms, how can they do so?

The coin placement problem: You are playing a game on a perfectly round table, in which both you and your opponent have access to an unlimited supply of equal-sized coins. Players take turns placing coins on the table, such that no other coins are disturbed, and each coin sits flat on the table (you are not allowed to place a coin on top of another coin). The last player who succeeds in placing a coin on the table without violating the previous constraint wins the game. Assuming you play first, what strategy guarantees you a win?

The 100 light bulbs: You have 100 toggle switches, each connected to a corresponding light bulb. At first, all lights are off. You then press every single toggle, thus turning on all of the bulbs. Next, you press every second toggle (ie. 2, 4, 6, 8, ..., 96, 98, 100) so that all the even numbered bulbs are turned off. Then you press every third toggle (3, 6, 9, ... 93, 96, 99) and every fourth toggle (4, 8, 12, ... 92, 96, 100), and every fifth toggle (5, 10, 15, ... 90, 95, 100), and so on, and so on, until the last two sequences, which are every ninety ninth toggle (99) and every hundredth toggle (100). Question: Which light bulbs will now be on?