# Riddles

With a background in logic and math, I love riddles. All of the riddles here rely on math and/or logical thinking rather than word tricks. No answers posted here, anybody reading this is free to email me though.

Thanks to many friends who've shared these with me and allowed me to post here.

### Bowling ball in the pool

**Difficulty:** medium

You're sitting in a rowboat in a swimming pool with a bowling ball on board. You throw the bowling ball out and it sinks to the bottom of the pool. Does the water level at the walls of the pool go up, go down or stay the same?

### Martini glass

**Difficulty:** easy

If you're drinking from a glass that is exactly an (upside-down) cone, by what ratio does the depth of the liquid decrease when you have consumed half of it?

### Jelly bean jar

**Difficulty:** easy

You're given a large jar of black and white jelly beans. You repeatedly reach in and pull out two random beans. If you pull out two beans of the same color, you put one white bean back in. If they are a different color, you put one black bean back in. Assume you always have enough extras sitting on the table to put one of either color back in.

As you repeat this process, eventually there is only one bean left in the jar. What color will it be? More specifically, what is the minimum amount of information you will need about the contents of the jar in order to determine the color of the final bean?

### Nuclear vault

**Difficulty:** easy

6 scientists are collaborating on experiments for which they need access to a vault with highly enriched uranium. However, none of the scientists is considered trustworthy. The scientists agree that if a majority of them (4 or more) are present, they should be able to open the vault, but 3 or fewer should not be able to. To achieve this, they can use as many ordinary locks as they want, and for each lock can make as many copies of the key as they want. The solution will involve using a number of locks, and giving each lock's keys to some subset of the scientists.

How many locks must be used so that no group of three scientists can unlock all of them, but any group of four or more can? How are keys for the locks distributed?

### Alligator lake

**Difficulty:** medium

You're at the center of a perfectly circular lake of radius 1. There is a man-eating alligator at the edge of the lake that is unwilling to enter the water but is capable of running *K* times faster than you can swim. Once you reach land, you can outrun the alligator as long as it is not directly at the point you hit the shore. What is the largest value of *K* for which you can safely exit the water without being eaten?

*Bonus:* What is the minimum distance you need to swim as a (piecewise) function of *K*?

### Monastic suicide

**Difficulty:** medium

100 monks live in a monastery. They follow a strict code of no communication, verbal or otherwise. One night they are all told that an unknown non-zero number of them have not been faithful and must commit suicide. The doomed monks will wake up the next morning with an 'X' on their foreheads.

Unfortunately, they have no mirrors and thus no way to know if they are marked. They do, however, all gather for dinner each night, at which time they can see which of the other monks is marked. Of course, they cannot communicate this. The doomed monks must kill themselves in their room at night.

How can the doomed monks figure out that they are marked? After how many days will all of them have killed themselves?

### Wandering polar bear

**Difficulty:** medium

A polar bear travels 1 mile south, 1 mile west, and 1 mile north. After this journey she's back where she started. What is the set of all possible locations the bear might have started/finished? Assume Earth is a perfect sphere.

### Three princesses

**Difficulty:** medium

A prince has a choice of three princessess to marry who are all sisters. The oldest always tells the truth. The youngest always lies. The middle princess sometimes tells the truth and sometimes lies. The prince wants to marry either the oldest or youngest sister, realizing that he can simply negate everything the youngest says and know the truth.

However, the king will only allow him to ask one yes/no question to one of the sisters, after which he must pick. The sisters all know each other's truth-telling patterns. What one question can he ask to a princess in order to guarantee he doesn't end up with the middle princess?

*Note:* while there are many possible complicated solutions, it is possible to ask only a simple question that a five-year old could understand and answer.

### 12 coins, 3 weighings

**Difficulty:** hard

A banker receives 12 coins, one of which is counterfeit and might be heavier or lighter than the 11 legitimate coins. Using a balance scale, and exactly 3 weighings, he can determine exactly which coin is counterfeit, and whether it is heavier or lighter than the good coins.

How is this done? A solution should include a decision tree of exactly which coins are weighed at each step, contingent on the previous steps. Unfortunately, the solution does not have a simple description.

This riddle can be generalized to N total coins, C counterfeit coins, but as far as I know there isn't an elegant solution for the minimum numbe of weighings.

### Warden's lamps

**Difficulty:** hard

11 prisoners are being held by a malicious warden. He offers them a game to earn their freedom. In an otherwise empty room, he has two lamps. The warden will repeatedly pick inmates as he pleases and take them to the lamp room, upon which they must switch on or off exactly one of the lamps.

The prisoners are allowed to meet for one hour to discuss strategy, but then they will be solitarily confined in their cells and have no communication except through the lamps. Their goal is to determine when each inmate has been to the lamp room. If one prisoner can say this with certainty upon entering the lamp room, the prisoners will all be freed. There is no penalty for making this claim late as long as it is correct. However, they will all be executed if the claim is made too early.

The warden can observe the prisoner's strategy meeting, and thus can take the prisoners into the lamp room in any order which he thinks will foil their strategy. He is also free to set the lamps on or off initialy as he wishes, although after that he will not change the lamps himself.

What strategy can the prisoners use to eventually state with certainty that all have visited the lamp room?

*Bonus:* If you solve this and I can send you a extra bonus problem (the stating of which gives away part of the answer to the original, so I won't post it).