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Index of Challenge Problems
- 1,234 to the 23 rd Power Title of Problem Page
- 2 Trains Meet
- ABCDEF Equations
- Alicia’s Babysitting Job
- Angelina’s Club
- Average of a List of 7 Numbers
- Basketball Bounce
- Best Athlete of the Day
- Bikes and Trikes
- Bobby’s Mariners Tickets
- Brei’s Long-Distance Call
- Checkerboard Problem
- Class President
- Comparing Ages/ Collin and Anthony
- Comparing Weights of 4th Graders
- Courtney’s Rectangular Blocks
- Dice Game
- Father and Sons Crossing Lake
- How Many Handshakes?
- How Many Squares/Rectangles?
- How Many Triangles?
- Jeremy’s Hondas
- Magic Number Box
- Making Whole Numbers from 2, 4, 6, and
- Monica’s Square Blocks
- Mr. Mayer’s Shirts & Ties
- Planet Grumble 1000-Day War
- Rectangular Patterns
- Restaurant Tables
- Sam the Soccer Man
- Shape Equations
- Sliding Slug on a Slanted Sidewalk
- Soccer Teams
- Stacey’s Schedule
- Tennis Tournament
- Tommy’s 12-Hour Clock
- Triangular Patterns
- What Number Am I?
- Work Schedules
- XYZ Equations
1234 to the 23 rd
Name ____________________________________ Date _______________
The Problem
(1,234)² means 1,234 * 1,234; (1,234)³ means 1,234 * 1,234 * 1,234; and so forth. When (1,234)²³ is completely multiplied out, what will the number be in the ones place?
2 Trains Meet
Name ___________________________________ Date _________________
The Problem
A train leaves Rock City at an average speed of 50 miles per hour and heads for Gnome City. Another train leaves Gnome City at an average speed of 40 miles per hour and heads for Rock City. If the route is 360 miles long, how many hours will it take for the 2 trains to meet?
2 Trains Meet (continued)
Solution Strategy: Draw a Picture/Guess and Check
Let’s first draw a picture to make the problem a little clearer.
360 miles
Rock City Gnome City Train 1 Train 2
Look at the picture carefully. We notice from the picture that the distance traveled by Train 1 up until the moment the 2 trains meet added to the distance traveled by Train 2 up until the moment the 2 trains meet is equal to the total distance between the 2 cities.
Now let’s make some guesses about the solution. Let’s say that the 2 trains would meet after 2 hours. Let’s try it.
Train 1 is traveling at 50 mph Æ for 2 hours Æ that’s 50 x 2 or 100 miles Train 2 is traveling at 40 mph Æ for 2 hours Æ that’s 40 x 2 or 80 miles 100 miles + 80 miles = 180 miles
No. We were told that the distance is 360 miles. Our guess is about ½ as big as it needs to be.
Let’s try 4 hours.
Train 1 is traveling at 50 mph Æ for 4 hours Æ that’s 50 x 4 or 200 miles Train 2 is traveling at 40 mph Æ for 4 hours Æ that’s 40 x 4 or 160 miles 200 miles + 160 miles = 360 miles
That’s it!
ANSWER: It will take 4 hours for the 2 trains to meet.
ABCDEF Equations (continued)
Solution Strategy: Look for a Pattern
Let’s first look at the equations and see if we recognize any of the patterns. As we look through the list, we see the equation B + C = B. We know that when 0 is added to a number, it stays the same. Therefore, C must be equal to 0. C = 0
Let’s look at the other equations for a pattern we recognize. We see D * E = D. We know that when a number is multiplied by 1, it stays the same. E = 1
Solution Strategy: Guess and Check
Are there any other equations that have familiar patterns or combinations? None really stand out. So we need to try another strategy. Let’s make some guesses at values and try them out. The equation with all A’s is interesting. What digit tripled is the same as that digit squared? We know that it must be a relatively small number because when you start squaring numbers, they get large fairly quickly. So let’s focus on small numbers. We’ve already used 0 and 1. Let’s try 2. 2 + 2 + 2 = 6 Does 2² = 6? NO, it’s 4.
That wasn’t it. Let’s try 3. 3 + 3 + 3 = 9 Does 3² = 9? YES, that’s it! A = 3
We now know the values for A, C, and E. We see that there is an equation with 2 of the values, A – C = B. Let’s substitute in the values for A and C (3 – 1 = 2). B = 2 Now that we know the value for B, we can solve the equation, B² = D (2² = 4). D = 4
Now that we’ve solved for D, we can substitute the values for D and E into the equation D + E = F and solve for F (4 + 1 = 5). F = 5 (We might also notice that there is just one variable left and one digit left, so F has to be 5!) Now we have solved for all of the variables.
ANSWER: The values for the variables A through F are as follows:
A = 3, B = 2, C = 0, D = 4, E = 1, F = 5.
Alicia’s Babysitting Job
Name ____________________________________ Date _______________
The Problem
Alicia was paid $125 for babysitting five days after school for the Smith family. Each day Mrs. Smith paid her $3 more than the day before. How much money did she earn on the first day?
Angelina’s Club
Name ____________________________________ Date _______________
The Problem
Angelina and her friends formed a club named the Extremely Cool Club. They wanted to assign a unique 4-digit secret code number to each member of the club. They decided to use the digits 1, 3, 7, and 9 for their numbering system and each of these digits can appear only once in every secret code number (i.e. 1379 is a valid number, but 1133 is not a valid number). What is the maximum number of members who could join the club if everyone is to be assigned a unique secret code number?
Angelina’s Club (continued)
Solution Strategy: Make a List
Let’s make a list to show all the possible combinations of 4-digit numbers that can be made from the digits 1, 3, 7, and 9.
1379 1397 1739 1793 1937 1973 3179 3197 3719 3791 3917 3971 7139 7193 7319 7391 7913 7931 9137 9173 9317 9371 9713 9731
ANSWER: There are 24 different unique combinations of 4-digit secret
code numbers available for the club. They can have 24 members in their
club.
Average of a List of 7 Numbers (continued)
Solution Strategy: Draw a Diagram
First, let’s think about what we know about the term average. We know that when we are finding the average of a group of numbers, we add them up and find their sum, and then we divide that sum by how many numbers we have in the group. So if we are told that we have 7 numbers in the group and the average is 9, what do we know about the sum of the numbers in the group? Let’s draw a rate diagram.
Units Sum of all the numbers in the group
How many numbers in the group?
The average of the numbers in the group Numbers? 7 9
We find that the sum of the group of 7 numbers is 63. Now that we see this relationship, we can look at the next part of the problem. If one of the numbers is taken away – we had 7 numbers and now we have 6 – the average is 8. Now we are looking at a group of 6 numbers with an average of 8. Again, let’s draw a rate diagram.
Units Sum of all the numbers in the group
How many numbers in the group?
The average of the numbers in the group Numbers? 6 8
Now we know what the sum of the numbers were for each case in our problem, but have we solved the problem? No. We are trying to determine what number was taken away from the first group, to get an average of 8 in the second group. Therefore, we need to take our 2 sums and find the difference.
63 – 48 = 15
The difference between those 2 sums is 15. That means that the number removed from the first group of numbers was 15.
ANSWER: The number removed from the list of 7 numbers was 15.
Basketball Bounce
Name ____________________________________ Date _______________
The Problem
A basketball bounces back up about ½ the height from which it is dropped. If a basketball is dropped from 120 feet and keeps bouncing, what is its vertical height after it hits the floor for the 3 rd^ time?
Best Athlete of the Day
Name ____________________________________ Date _______________
The Problem
Bakersfield Elementary School has a field day at the end of the school year in which students compete in sporting events. A group of 5 friends wanted to determine who was the “best” athlete of the day based on the results of 3 events – the 50-yard dash, the mile run, and the rope climb. The chart below shows their results in the 3 events. Each of the events is of equal importance. Tell who you think is the “best” athlete of the day based on the results shown in the chart. Explain why you chose that person.
Names 50-yard Dash Mile Run Rope Climb Brei 8 seconds 6.12 minutes 6 feet Charlie 10 seconds 8 minutes 7.5 feet Dani 7.5 seconds 12 minutes 5.8 feet Eddie 9 seconds 5.59 minutes 4.3 feet Francine 12.2 seconds 10 minutes 8.2 feet
Best Athlete of the Day (continued)
Solution Strategy: Make a Simpler Problem
Let’s take another look at the chart:
Names 50-yard Dash Mile Run Rope Climb Brei 8 seconds 6.12 minutes 6 feet Charlie 10 seconds 8 minutes 7.5 feet Dani 7.5 seconds 12 minutes 5.8 feet Eddie 9 seconds 5.59 minutes 4.3 feet Francine 12.2 seconds 10 minutes 8.2 feet
It’s difficult to make comparisons between the 3 events because they are measured in different units. The 50-yard dash is measured in seconds; the mile run is measured in minutes; and the rope climb is measured in feet. It is also important to note that the lower values are the better scores in the 50-yard dash and the mile run, but the higher values are the better scores in the rope climb. How can we make the problem simpler?
Let’s assign a number to each person that shows their ranking in each event. Since there are 5 people, let’s rank them 1 – 5 in each event with a 5 being the best score and a 1 being the worst score. Our chart would look like this:
Names 50-yard Dash Ranking
Mile Run Ranking
Rope Climb Ranking
Total Score Brei 4 4 3 11 Charlie 2 3 4 9 Dani 5 1 2 8 Eddie 3 5 1 9 Francine 1 2 5 8
We see that Brei has a score of 11 which is the highest score of the five friends. Therefore, we would choose Brei as the overall “best” athlete for these 3 events.
ANSWER: Brei is the best athlete of the day.
Bikes & Trikes (continued)
Solution Strategy: Guess and Check
We know that bicycles have 2 wheels and tricycles have 3 wheels. We know that there are a total of 12 bicycles and tricycles at the park. Let’s make some guesses.
Let’s say there are 6 bicycles and 6 tricycles.
6 * 3 = 18 wheels 6 * 2 = 12 wheels
18 + 12 = 30 wheels
No, that’s too high. But only by 1. So we know that we are close. Let’s try adjusting the numbers just slightly. Let’s say there are 7 tricycles and 5 bicycles.
7 * 3 = 21 wheels 5 * 2 = 10 wheels
21 + 10 = 31 wheels
No. This is even higher. It looks like we adjusted our guess the wrong way. Let’s try 7 bicycles and 5 tricycles.
7 * 2 = 14 wheels 5 * 3 = 15 wheels
14 + 15 = 29 wheels
That’s it! We have found the combination of 12 bicycles and tricycles that gives us 29 wheels. It is 7 bicycles and 5 tricycles.
ANSWER: There are 7 bicycles and 5 tricycles at the park.
Bobby’s Mariners Tickets
Name ____________________________________ Date _______________
The Problem
Bobby has 4 tickets for the Mariners game. He invites 3 friends – Tommy, Larry, and Sam – to go with him. Bobby has the first 2 seats in row 5, and the first 2 seats in row 6. The boys are trying to decide on a seating arrangement. How many different combinations of seating arrangements can the boys choose from?
