**Some of My Favorite
**

**Allen Arnold, April 2003 **

(Hint: Cover the answers with a piece of paper so you don’t see them as you work on the puzzles )

**Draw one straight line to make the equation true.**

**If it takes 15 minutes to cut a log into 2 pieces, how long does it take to cut it into 4 pieces?**

(It’s not a proportion, so the answer is 45 minutes, not 30 minutes)

**If you have 9 red socks and 12 blue socks all mixed up in a drawer, how many would you have to select (without looking) to be sure you had a matching pair?**

**Suppose 6 people shake hands with each other. How many handshakes are there?**

**A bottle and a cork together cost $1.50. If the bottle costs a dollar more than the cork, how much does the cork cost?**

**If you read a book starting at page 100 and finish on page 200, how many pages did you read?**

**A cube-shaped piece of cake is covered with chocolate icing on all sides. The cake is cut as shown into 27 small cubes. How many of the small cubes are covered with icing on three sides?**

(Only the 8 corner cubes have icing on three sides)

**What fraction is halfway between one-third and one-fourth?**

(1/3 equals 8/24 and 1/4 equals 6/24, so 7/24 is exactly halfway between.)

**Robert is three times as old as Susan, but in two years he’ll only be twice as old as Susan. How is this possible?**

(Robert is 6 years old and Susan is 2 years old, so right now he’s three times as old. But in two years Robert will be 8 and Susan will be 4, so he’ll only be twice as old.) NOTE: Robert will *always* be 4 years older than Susan.

**Suppose you need to measure exactly 4 quarts of oil for your car. You only have a 3 quart can, a 5 quart can, and a big tank of oil. Without estimating, how can you measure exactly 4 quarts?**

(Fill the 3 quart can and pour it into the 5 quart can. Fill the 3 quart can again and pour as much as you can (2 quarts) into the 5 quart can. Now the 3 quart can has 1 quart in it and the 5 quart can is full. Empty the 5 quart can into the tank. Pour the 1 quart from the 3 quart can into the 5 quart can. Fill the 3 quart can again and pour it into the 5 quart can. That makes 4 quarts in the 5 quart can)

**Your scores on math tests so far this year are 50, 60, 70, and 80. What score would you need on the next test to have an average of 70?**

(You would have to score a 90)

**Suppose a large sheet of paper, one-thousandth of an inch thick is torn in half and the two pieces put together, one on top of the other. These are then torn in half and the four pieces put together in a pile. If this process of tearing and piling is repeated 50 times, will the final pile of paper be more or less than a mile high?**

(It would be almost 18 *million* miles high!)

**Suppose you have ten stacks of coins and each coin weighs 1 gram, except one stack contains coins that weigh 1.1 grams each. You don't know which stack contains the heavier coins. In ONE weighing how can you tell which stack contains the heavier coins? **

Put one coin from stack #1 on the scale. Put two coins from stack #2 on the scale. Continue in this way until you put ten coins from stack #10 onto the scale. If all the coins weighed the same, the total weight would be exactly 55.0 grams, since there are 55 coins on the scale. But the weight will be more than this because some of the coins are heavier. So how can you tell which stack has the heavier coins? Suppose the total weight is 55.6 grams. This would be 0.6 grams more than if the coins weighed the same. So there must be 6 of the heavier coins and they could only have come from stack #6. I love this problem!

**Suppose you buy 100 pounds of cucumbers and they are 99% water. After a few days, they dry out to 98% water. How much do the cucumbers weigh now? The answer will surprise you! **

Solution:

99 lbs water / 100lbs = .99 (originally)

Let x = pounds water lost by evaporation, then:

(99 - x) / (100 - x) = .98 (after evaporation of x pounds water)

Solving for x, we get x = 50 pounds, so there's only 50 pounds left!!

**Which is bigger: 99^100 + 99^100 or 100^100? This can be done easily with logarithms, but it's more "elegant" and more fun to do it without them. **

You're on your own here.

**In 1999 a small forest contains 4000 trees. Each year, 20% of the trees are harvested and 1000 new trees are planted. Will the number of trees always increase? Will the number of trees ever decrease until the forest "dies out"? If so, how long will that take? Will the number of trees "stabilize" after some number of years? If so, when? How many trees would there be at that time? Is there a formula that tells the number of trees, given the number of years? **

You're on your own here too.

**There is a very good reason why manhole covers are round. The reason is based on a simple geometric property of circles. So, why are manhole covers round?**

Here too.

**Which is better, a discount of 15% or a discount of 10% followed by a discount of 5%? There's a difference.**

**A car travels from point P to point Q at 40 mph, and then from Q to P at 60 mph. What's the average rate for the round trip?**

It's not 50 mph!

**You're walking on a path and come to a fork in the road. One fork leads to a dead end. The other is the correct fork. You don't know which way to go. There is a Truth-Teller and a Liar standing at the fork. You don't know who is who. What question can you ask to find out which is the correct path? **

Ask either one what the other would say if asked "Is the left fork the correct path?"

**A group of 64 kids compete against each other in a math tournament. Each kid is paired with another and each pair is given a math problem to solve. The winner stays in the tournament and the loser is eliminated, so each time, one half of the participants is eliminated and the remaining half forms pairs again. How many problems altogether must be solved to declare a winner?**

In order to declare a winner, 63 kids must be eliminated, so there are 63 problems. Another way is to add up the eliminations at each stage of the tourament: 32 + 16 + 8 + 4 + 2 + 1 = 63.** **