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# Problem of the Month Archive

## March 2003

### How heavy is a brick?

If a brick weighs the same as three-quarters of a brick plus 3/4 pound, how much does a brick weigh?

Solution

### Crossing bridges in the dark

Four travelers, Aaron, Bill, Chuck, and Dave, must cross a bridge in the dark. Aaron can cross in 10 minutes, Bill takes 7 minutes, Chuck 3 minutes, and Dave can do it in 1 minute. The bridge is very fragile and can hold at most two travelers at a time. Because it is dark, crossing the bridge requires a flashlight but amongst them they only have one flashlight. When two people cross at the same time they can share the flashlight. The problem is that the bridge is falling apart and will be unsafe to use in 20 minutes. How can all four travelers safely cross the bridge in the next 20 minutes?

Solution

## April 2003

### How many ways can you spell FUN MATH?

Starting with the "F" at the top of the triangle, following the lines and moving down letter by letter, how many ways can you spell FUN MATH?
 Here is an example of one way:
Solution

### A hiking we will go...

Jeff and Jan both walk at the same speed. On flat ground they walk at 3 miles per hour. They walk uphill at 2 miles per hour and downhill at 4 miles per hour. They are going on a hike where there are two paths to the destination. One is entirely flat, while the other is for uphill half of the distance and downhill for half of the distance. The total distance of each path is the same. Jeff takes the flat path while Jan takes the uphill-downhill path. Who arrives at the destination first?

Solution

## May 2003

### Nuts and bolts

Sue and Tom went to the hardware store to buy nuts and bolts for some household repairs. Sue bought a bag of bolts and it cost her \$1.54. Tom bought a bag of nuts and it cost him \$0.99. They each bought the same number of items. How many nuts and bolts did they buy, and what did each cost?

Solution

### Breaking up a dozen

You have 12 identical balls. Are there more ways

1. to divide up the balls into 3 groups of any sizes

or

2. to divide up the balls into any number of groups the largest of which has exactly 3 balls in it?

Solution

## June 2003

### Fast Clock

A 12-hour wall clock runs too fast. Every hour it gains 5 minutes. For example, if it shows the correct time at 12:00 noon, at 1:00 PM, it will display 1:05. If it shows the correct time at noon on Sunday June 1, 2003, at what date and time will the clock next show the correct time?
Solution

### Fast Clock - Slow Clock

You have two 12-hour wall clocks. One runs fast by 2 minutes per hour, while the other runs slow by 3 minutes per hour. If you set the slow clock, for example, to the correct time at 12:00 noon, at 1:00 PM it will show 12:57. You set both clocks to the correct time at noon on Sunday June 1, 2003. At what day and time will the two clocks next display the same time?
Solution

## July 2003

### Black and Blue Hands

The clock at the right has a blue hour hand and a black minute hand. At noon the hands are exactly aligned to point to the 12. How many more times will the hands be aligned again before midnight?
Solution

### One Hand Times

This clock has two black hands. When they point to the same place on the clock it looks like the clock has only one hand. What time (exact) does this one-handed clock show?
Solution

## August 2003

### Colored Socks

John has 50 socks in a drawer. The colors of socks are red, blue, or green. John says, "No matter how you choose three socks from the drawer, you will get at least one red sock." Assuming that there is at least one sock of each color, how many of each type of sock does John have?
Solution

### Starred Socks

Jenny keeps her pairs of socks in a drawer. The socks are decorated with stars and each pair of socks has the same number of stars. Every sock has at least one star but socks can have a different number of stars (even two socks in a pair might have a different number of stars). Jenny says to you, "I have more than one pair of socks. Altogether, there are between 100 and 150 stars. Now that you know this you can tell me how many socks I have!" How many socks does Jenny have in her drawer?
Solution

## September 2003

### Tennis Tournament

Six players, Ann, Beth, Cheryl, Diane, Erica, and Faye, compete in a tennis tournament. Each player plays one match against each of the other players. How many tennis matches will this tournament have?
Solution

### Tournament Week

In the tournament of the previous problem, each player can compete in only one match per day. Is it possible for them to complete the tournament in five days? How?
Solution

## October 2003

### Lines

Nine gold coins are arranged in a square pattern as shown to the right. There are many lines formed by two or three of the coins. Line 1, for example, has three coins on it, while line 2 has only two coins. How many lines can you find that contain exactly two coins? How many lines can you find that contain exactly three coins?
Solution

### Two More Lines

The nine coins from the previous problem can be rearranged so that the number of lines containing exactly three coins is increased by two! Show how.

Solution

## November 2003

### 20 Castle guards

24 guards are positioned along the walls of a square castle in groups of 3 as shown on the right. In this way, each side of the castle has a total of 9 guards protecting it. During the enemy's last raid, 4 of the guards are killed. How can the remaining 20 guards be placed around the castle walls so that each side of the castle still has 9 guards protecting it?
Solution

### 18 Castle guards

2 of the 20 surviving guards become ill and are unable to help guard the castle. How can the remaining 18 guards be placed around the castle walls so that each side of the castle still has 9 guards protecting it?
Solution

## December 2003

### Train Lines

Fanoland has seven cities: Agra, Basil, Canon, Darma, Erina, Foxie, and Goren. The Fanoland railroad is building a network of train lines. Each train line goes through exactly three cities. Each city in Fanoland is connected to every other city by exactly one train line. For example, the yellow train line built through Agra, Basil, and Canon connects Agra and Basil, Basil and Canon, and Canon and Agra. How many train lines will go through each city?
Solution

### Train Network

Show a network of train lines that satisfies the conditions above.
Solution