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Problem of the Month Archive 
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Problems from Previous Years
 
Problems from 2003

How heavy is a brick?If a brick weighs the same as threequarters 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 
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? 



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 uphilldownhill path. Who arrives at the destination first? Solution 
Nuts and boltsSue 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 or2. to divide up the balls into any number of groups the largest of which has exactly 3 balls in it? Solution 
Fast ClockA 12hour 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 12hour 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 
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 onehanded clock show?  
Solution 
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 
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 
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 
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 
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 
 
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