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.

The Lily Pad

A lily pad starts out very small, but doubles in size every day. After 60 days it has completely covered a pond. After how many days had it covered one-quarter the area of the pond? Click below for the answer.

The instinctive answer might seem to be 15 days, but that would only be correct if the lily pad was growing linearly. Remember, the lily pad doubles in size every day, which is exponential. To solve this puzzle, just work backwards. If the lily pad completely covers the pond on day 60, then half the pond was covered on day 59, and one-quarter of the pond was covered on day 58.