Roger's Favorite Puzzles

I like puzzles. Below are some of my all time favorites. You will get additional information (remarks, hints, solution) if you hit the puzzle name. If you have comments or suggestions for other puzzles, please drop me an email.

With regards/thanks to Gurmeet Singh Manku. Not only did I steal his layout, but also some puzzles of his collection. Thanks to Peter Winkler, Rustan Leino and Michael Brand for some other puzzles. Thanks to Barbara Keller, Fabian Kuhn, Thomas Locher, Mike Paterson for yet some other puzzles.

Two Minute Puzzles

The Parking Lot: What is the number of the parking space containing the car? (Please hit the picture for a larger version.)

Cheryl's Birthday: Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17. Cheryl tells Albert the month, and Bernard the day. First Albert says: "I don't know Cheryl's birthday, but Bernard does not know as well!". Bernard then says: "Now I know Cheryl's birthday, and so does Albert!". When is her birthday?

Forty-five Minutes: How do we measure forty-five minutes using two wires, each of which takes an hour to burn. Unfortunately, the wires burn non-uniformly (sometimes faster, sometimes slower). We have plenty of matchsticks.

The Blind Man: A blind man was handed a deck of 52 cards with exactly 10 cards facing up. How can he divide it into two piles, each of which having the same number of cards facing up?

The Rope: You are trapped atop a 200m high building. You have a rope 150m long, plus a Swiss Army knife. There is a hook at the top where you stand. Looking down, at midway between you and the ground, at a height of 100m, there is a ledge with another hook. Is it possible to get down safely?

Duck and Fox: A duck swims in a perfectly circular pond. There is a fox at the shore, afraid of water, that plans to catch the duck when getting out. The land speed of the fox is four times as high as the water speed of the duck, however, once the duck reaches the shore without the fox in its immediate neighborhood, it can hide and escape. Can the duck reach the shore safely?

The Triangles: Both "triangles" are made of exactly the same pieces. So where is the gap in the lower triangle coming from?! (Please hit the picture for a larger version.)

99 Cops: A town has 99 cops. A cop is either honest or corrupt, the majority of the cops is honest. You need to figure out all the corrupt cops, with less than 299 questions. All cops know who is honest and who is corrupt, but only honest cops will answer truthfully. Corrupt cops may lie arbitrarily. For security reasons you can only ask one type of question: You may ask cop X whether cop Y is corrupt. This question will by answered by X with either "Y is corrupt" or "Y is honest".

That's impossible! Or is it?!?

Two Cards: You have to play a series of 1000 games against an opponent, and win at least 501 of them. In each game, the opponent chooses two carefully selected cards from the 13 hearts cards from the deck, from 2 up to ace. You can sample one of the two cards, and must then decide whether the still hidden card is higher than the card that was revealed. If you are right, you win the game. How can you make (almost) sure to win 501 out of 1000 games?

12 Men: In episode 18 of season 2 of TV show Brooklyn Nine-Nine, Captain Ray Holt challenges his crew with the following puzzle: There are twelve men on an island, eleven weigh exactly the same amount, but one of them is slightly lighter or heavier, you must figure out which. The island has no scales, but there is a see-saw. The exciting catch: You can only use it three times.

Four Cards: You have to win a game against an opponent. Before the game starts you are blindfolded. There are four cards placed on a square table, one card at each corner. The initial configuration of the cards is chosen by the opponent, arbitrarily and unknown to you. Your goal is to have all four cards face up. In each move you can select any subsets of the four cards, which are then flipped simultaneously by the opponent. After your move, if all four cards are face up, you win. If not, the opponent may rotate the table by an amount of his choice (90, 180, 270, or 360 degrees). If you don't manage to have all four cards face up in 20 moves or less, you lose. What's your strategy?

100 Hats: To be released from prison, 100 prisoners have to win a game against the hangman. The hangman gives a hat to every prisoner, the hat shows a two-digit integer number between 00 and 99. The numbers do not have to be different, the hangman can also choose to give each prisoner a hat with the same number. Prisoners can see the numbers on the other hats, but they cannot see their own number. Before playing the game, prisoners can discuss their strategy, but once the game starts there is an absolute communication stop. Now each prisoner has to guess his number. If any prisoner guesses his number correctly, all prisoners are released. If no prisoner guesses right, the hangman executes his job. What's the strategy?

The Switch: The hangman summons his 100 prisoners, announcing that they may meet to plan a strategy, but will then be put in isolated cells, with no communication. He explains that he has set up a switch room which contains a single switch, which is either on or off. It is not known to the prisoners whether the switch initially is on or off. Also, the switch is not connected to anything, but a prisoner entering the room may see whether the switch is on or off (because the switch is up or down). Every once in a while, the hangman will let one arbitrary prisoner into the switch room. The prisoner may throw the switch (on to off, or vice versa), or leave the switch unchanged. Nobody but the prisoners will ever enter the switch room. The hangman promises to let any prisoner enter the room from time to time, arbitrarily often. That is, eventually, each prisoner has been in the room at least once, twice, a thousand times, any number you want. At any time, any prisoner may declare "We have all visited the switch room at least once". If the claim is correct, all prisoners will be released. If the claim is wrong, the hangman will execute his job (on all the prisoners). What's the strategy?

100 Boxes: To be released from prison, 100 prisoners have to win a game against the hangman. The names of 100 prisoners are placed in 100 wooden boxes, one name per box, and the boxes are lined up in a room. One by one, the prisoners are led into the room; each may look into at most 61 boxes (61 is the lucky number of the hangman!), but must leave the room exactly as he found it and is permitted no further communication with the others. The prisoners have a chance to plot their strategy in advance. If each and every prisoner finds his name, all prisoners are released. If only one prisoner does not find his name, the hangman executes his job (on all the prisoners). Is it possible to come up with a scheme such that the chance of winning is higher than losing?

The Village: Many years ago there was a tiny village, populated with 20 couples. The husbands meet every noon to discuss village matters. The husbands all were horribly jealous. If one ever figures out that his wife ever cheated on him, he will kill himself the following night, at midnight. Obviously, no (living) husband knew whether his wife cheated. Despite all this, gossiping was popular in the village, and every husband knew exactly which wives cheated (apart from his own wife of course). One day an ethnologist came to town. After a few days in the village he announced he was surprised to find cheating (in such a small village). After that the ethnologist left town again. What happened next?

The Number: You must guess the secret number of a mathematical Rumpelstiltskin. For all statements x below: If statement x is true, then x must be a digit of the secret number; if statement x is false, the digit x must not be in the number. What's the secret number?
1. The product of the digits is odd.
2. Every digit is smaller than the next digit (if available).
3. No two digits are the same.
4. No digit is larger than 4.
5. The number has less than 6 digits.
6. The product of the digits is not divisible by 6.
7. The number is even.
8. No two digits have a difference of 1.
9. We have a = b + c, where a, b, and c are three digits of the number. (If a digit is used twice in the number, we can also use it twice.)
0. This number does not exist!

The Queue: 100 men stand in a queue, all looking in the same direction. Each man is wearing a hat. Hence, the last man can see all hats but his own, while the first man in the queue cannot see anybody's hat. The hats have a shape of a single digit, from 0 to 9. Starting with the last man in the queue, each man says a single digit, hopefully the digit on its own hat. Show how almost all the men can guess correctly!