True or False?

You're taking your college entrance exam, and things are not going well. You spent too much time answering the essay questions and word problems, and now you're running out of time. You turn to the final section and are relieved to find that it consists of 100 True/False questions. The bad news is that you only have 2 minutes left to complete the entire exam!

You figure it doesn't matter if you answer all True, all False, or just guess randomly. Odds are you'll probably get half of them right either way. You begin scribbling in your answers.

How many different ways (unique sequences) are there to answer 100 True/False questions? Click below for the answer.

If you're a computer programmer, you may have recognized right away that this is just a binary counting problem in disguise. Substitute 1 for True and 0 for False, and this is the same as asking how many unique values can be represented by 100 binary bits. Since that's every value from all 0s to all 1s, the answer is $2^{100}$, or 1,267,650,600,228,229,401,496,703,205,376.

King's Sentence

A man is caught poaching on the King's property. He is brought before the King to be sentenced.

The King says, "You must give me one statement. If it is true, you will killed by lions. If it is false, you will be killed by trampling of wild buffalo."

In the end, the King has to let the man go.

What was the man's statement? Click below to see the answer.

The man can get out of the punishment by making a paradoxical statement, such as "I will NOT be killed by lions." If that statement was true, the man would be killed by lions, making the statement false. Another statement that would have worked is "I will be killed by trampling of wild buffalo."

A Boy and a Girl

Two children, one boy and one girl, make the following statements.

"I am a boy" said the child with brown hair.

"I am a girl" said the child with red hair.

At least one of the children is lying. Who is the boy and who is the girl? Click below for the answer.

Both children are lying.

The child with the brown hair is the girl, and the child with the red hair is the boy.

(If only one child had lied, they would both be boys or both be girls.)

Number System

In a slightly eccentric numbering system, the numbers on the left are converted to regular decimal numbers by applying a simple rule.

9999 = 4
8888 = 8
1816 = 3
1212 = 0

Can you answer

2419 = ?

Click below for the answer.

To convert numbers from this system to decimal, just count the number of closed areas in each digit. The digit 9 has one closed area, so 9999 = 4. An 8 has two closed areas, so 8888 = 8, and so on. The digits in 2419 have two closed areas, so 2419 = 2.

Fraction of 1000

What is 1/2 of 2/3 of 3/4 of 4/5 of 5/6 of 6/7 of 7/8 of 8/9 of 9/10 of 1000? Click below for the answer.

At first glance, this problem looks a lot harder than it is. If you work backwards starting at 9/10 of 1000, it's easier to see that the final answer is 100.

Three Water Bottles

You have three water bottles with capacities of 8 quarts, 5 quarts, and 3 quarts. The largest bottle is filled with water, and the other two are empty. If there are no graduation marks on any of the bottles, how can you split the water evenly so that two of the bottles contain exactly 4 quarts each? You can only use these three bottles. Click below for the answer.

There may be other ways to solve this problem, but here's one sequence that works.

1. Fill the 5 quart bottle, leaving 3 quarts in the 8 quart bottle.
2. Pour 3 quarts from the 5 quart bottle into the 3 quart bottle, leaving 2 quarts in the 5 quart bottle.
3. Empty the 3 quart bottle into the 8 quart bottle , leaving 6 quarts in the 8 quart bottle.
4. Pour the 2 quarts from the 5 quart bottle into the 3 quart bottle.
5. Fill the 5 quart bottle from the 8 quart bottle , leaving 1 quart in the 8 quart bottle.
6. Pour 1 quart from the 5 quart bottle into the 3 quart bottle (filling it), leaving 4 quarts in the 5 quart bottle.
7. Pour the 3 quarts from the 3 quart bottle into the 8 quart bottle, leaving 4 quarts in the 8 quart bottle.

That's a lot to follow, so here are the steps in tabular form.

8 qt.	5 qt.	3 qt.
8	0	0
3	5	0
3	2	3
6	2	0
6	0	2
1	5	2
1	4	3
4	4	0

The Compulsive Gambler

You are approached by a compulsive gambler with the following proposal. You are to flip a fair coin four times. If heads and tails both appear twice each, he will pay you $11. If any other combination of heads and tails appears, you have to pay him only $10. Do you take the wager? Click below for the answer.

This is not a good gamble. Below are all the possible outcomes for four successive coin flips.

Notice that exactly two heads and two tails only appear six out of sixteen times, so you can only expect to win this game about 37.5% of the time. At the offered stakes ($11 for a win, $10 for a loss) you'd be losing an average of around $2.12 every time you play.

This problem appeared as an exercise in Introductory Graph Theory by Gary Chartrand.
See my Probability GitHub repository for a script that shows how to model this problem in Python.