## Saturday, December 16, 2017

### 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.

Labels:
logic puzzles,
math,
numbers

## Saturday, December 9, 2017

### 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.

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.

## Saturday, December 2, 2017

### 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**.## Saturday, November 25, 2017

### Three Coin Flips

The following game has a $10 entry fee. You are to flip a fair coin three times. The first time it comes up heads you are paid $5. The second time it comes up heads you're paid an additional $7. The third time it comes up heads you're paid $9 more, for a possible maximum prize of $21. Would you pay the $10 entry fee to play?

If not, what would be a fair price for this game? Click below for the answer.

It's not a good idea to play this game at the offered entry fee. Here are the eight possible outcomes when flipping a coin three times, along with how much you would win after each flip.

When you subtract out the $10 entry fee, you only win "big" ($11) one out of eight times. Three times you win only $2, three times you lose $5, and one out of the eight times you lose your entire $10 entry fee. If you average these, you can expect to lose $1 every time you play. So, if you lower the entry fee to $9 this would be a fair game.

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.

When you subtract out the $10 entry fee, you only win "big" ($11) one out of eight times. Three times you win only $2, three times you lose $5, and one out of the eight times you lose your entire $10 entry fee. If you average these, you can expect to lose $1 every time you play. So, if you lower the entry fee to $9 this would be a fair game.

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.

Labels:
math,
probability,
puzzles

## Saturday, November 18, 2017

### One equals 0.999...

The following is a mathematical proof that 1 is equal to 0.999.... What's wrong with it? Click below for the answer.

$x = 0.999...$

$10x = 9.999...$

$10x = 9 + 0.999...$

$10x = 9 + x$

$9x = 9$

$x = 1$

$10x = 9.999...$

$10x = 9 + 0.999...$

$10x = 9 + x$

$9x = 9$

$x = 1$

There's nothing wrong with it. 1 really is equal to 0.999...

## Saturday, November 11, 2017

### Two equals one?

The following is a mathematical proof that two equals one. What's wrong with it? Click below for the answer.

$a = b$

$aa = ab$

$aa - bb = ab - bb$

$(a + b)(a - b) = b(a - b)$

$a + b = b$

$a + a = a$

$2a = a$

$2 = 1$

$aa = ab$

$aa - bb = ab - bb$

$(a + b)(a - b) = b(a - b)$

$a + b = b$

$a + a = a$

$2a = a$

$2 = 1$

The problem is in the fourth step, where both sides of the equation are divided by $(a - b)$. Since $a = b$ is given at the start, $a - b$ is 0, and you can't divide by 0.

## Saturday, November 4, 2017

### State Names

There's only one letter in the English alphabet that is not used in the name of any of the 50 United States. Do you know which letter it is? Click below for the answer.

If you said 'J' or 'Z' you weren't far off. Each of those letters appear only once in the names of the 50 states (thanks to New Jersey and Arizona). The correct answer, though, is the letter

To find the solution, I used a Python script to load a list of state names and count the occurrence of each letter of the alphabet. This method takes a lot less time than consulting a map.

**Q**, which does not appear in any state name.To find the solution, I used a Python script to load a list of state names and count the occurrence of each letter of the alphabet. This method takes a lot less time than consulting a map.

Labels:
puzzles

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