Prisoner's Probability to Live

You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game. He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, "Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls and marbles up. You can then choose one bowl and remove one marble. If the marble is white, you live, but if the marble is black...you die.

How do you divide the marbles up so that you have the greatest probability of choosing a white marble?

Solution »

Place 1 white marble in one bowl, and place the rest of the marbles in the other bowl (49 whites, and 50 blacks).


This way you begin with a 50/50 chance of choosing the bowl with just one white marble and living. But even if you choose the other bowl, you still have an almost 50% chance of picking one of the 49 white marbles. There are no guarantees in life, but this is your best bet at surviving.

Three Daughters

I was visiting a friend one evening and remembered that he had three daughters. I asked him how old they were. "The product of their ages is 72," he answered. Quizzically, I asked, "Is there anything else you can tell me?" "Yes," he replied, "the sum of their ages is equal to the number of my house." I stepped outside to see what the house number was. Upon returning inside, I said to my host, "I'm sorry, but I still can't figure out their ages." He responded apologetically, "I'm sorry, I forgot to mention that my oldest daughter likes strawberry shortcake." With this information, I was able to determine all of their ages. How old is each daughter? You have enough information to solve the puzzle.

Solution »

3, 3, and 8. The only groups of 3 factors of 72 to have non-unique sums are 2, 6, 6 and 3, 3, 8 (both add up to 14). The presence of a single oldest child eliminates 2,6,6.

Bucket

You have two buckets - one holds exactly 5 gallons and the other 3 gallons. How can you measure 4 gallons of water into the 5 gallon bucket?

(Assume you have an unlimited supply of water and that there are no measurement markings of any kind on the buckets.)

Solution »

1. Fill the 3-gallon bucket.
2. Pour the 3 gallons of water into the 5-gallon bucket
3. Fill the 3-gallon bucket again.
4. Fill up the 5-gallon bucket with the 3-gallon bucket, leaving you with 1 gallon left in the 3-gallon bucket.
5. Empty out the 5-gallon bucket.
6. Pour the remaining 1 gallon of water from the 3-gallon bucket into the 5-gallon bucket.
7. Fill the 3-gallon bucket.
8. Pour the 3 gallons of water from the 3-gallon bucket into the 5-gallon bucket leaving you with 4 gallons of water in the 5-gallon bucket.

Alternate solution:

1. Fill up the 5 gallon bucket
2. Pour it into 3 gallon bucket, leaving 2 gallons
3. Empty out the 3 gallon bucket
4. Pour the 2 gallons in the 5 gallon bucket into the 3 gallon bucket
5. Fill up the 5 gallon bucket and pour it into the 3 gallon bucket until it's full, leaving 4 gallons in the 5 gallon bucket.

Sisters and Brothers

My daughter has many sisters. She has as many sisters as she has
brothers. Each of her brothers has twice as many sisters as brothers.
How many sons and daughters do I have?

Solution »

4 daughters and 3 sons.

For any particular daughter, she will have 3 sisters and 3 brothers. (DON'T count that daughter in -- she's the 4th girl -- otherwise it won't be 3 sisters = 3 brothers.)

But for any particular son, he will therefore have 4 sisters and 2 brothers. And 4 sisters are exactly twice as much as 2 brothers.

What is the next number in the sequence?

Here is a series of numbers. What is the next number in the sequence?

1
11
21
1211
111221
312211
13112221

Solution »

The next number in the sequence is 1113213211, because the rule for creating the next number is to simply describe the previous number. For example, you start of with 1, with is simply one 1, so the next number is 11. Now you have two 1's, so the next number is 21. Now you have one 2 and one 1, so the next number is 1211. The solution is to simply continue describing the previous number using only numbers.