Figure This! Math Challenge (Challenges 1-25)
Week 25
Question: If all grape juice concentrates are the same strength, which recipe would you expect to have the strongest grape taste?
Hint: For each recipe think about how much water should be used with 1 cup (c.) of concentrate, or how much concentrate should be used with 1 cup of water.
Week 24 Challenge Question
Question: The Environmental Protection Agency (EPA) estimates for gas mileage on 1999 cars vary widely. Which of the following cars should go the farthest on one tank of gas?
| Car | City Mileage | Highway Mileage | Fuel Tank Capacity |
| Century | 20 mpg | 29 mpg | 17.5 gal |
| Metro | 44 mpg | 49 mpg | 10.3 gal |
| Navigator | 13 mpg | 17 mpg | 30.0 gal |
Hint: Suppose you only drive in the city. On the highway. Miles per gallon is an example of a rate. Grocers, demographers, financiers, actuaries, and economists all use rates in their work.
Week 23 Challenge Question
Question: If each side of the triangle in Figure 1 is 1 inch long, this means the triangle has a perimeter of 3 inches. Suppose you continued the pattern in the diagram until you reached Figure 5. What is the sum of the perimeters of all the white triangles in Figure 5?
Hint: The perimeter of a shape is the distance around it. Find the sum of the perimeters for all the white triangles
in each of the figures above. What pattern do you see?
in each of the figures above. What pattern do you see?
Week 22 Challenge Question
Question: A combination lock uses three numbers from 0 to 39. It opens when these numbers are dialed in a particular order: right, left, right. How many possible combinations are there?
Hint: Think about how many choices you have before dialing each number.
Week 21 Challenge Question
Question: Some doctors use body-mass index as an indicator of health risk. According to The Old Farmer’s Almanac 2000, body-mass index (BMI) can be found using the formula:
BMI = [ (W x 705) ÷ H ] / H
where H is height in inches and W is weight in pounds. According to the Almanac, an index greater than 27 or less than 19 indicates an increased risk for health problems. Helix is 5 feet, 2 inches tall and weighs 110 pounds.
Hint: Convert Helix’s height to inches, then use the formula.
Week 20 Challenge Question
Question: There are four basketball games Saturday night. Three sportswriters predicted the winners in the Saturday morning paper.
• Perimeter picks the Raptors, Pacers, Magic, and 76ers.
• Exponent picks the Hawks, Pistons, Magic, and Raptors.
• Helix picks the Heat, Pacers, Pistons, and Raptors.
• No one picks the Bucks.
WHO PLAYED WHOM?
Hint: Teams chosen by the same sportswriter did not play each other.
Week 19 Challenge Question
Question: Your team is down by one point. Your teammate, who makes free throws about three-fourths of the time, is at the free throw line. She gets a second shot if she makes the first one. Each free throw that she makes is worth 1 point. If there is no time left, what are the chances you win the game with no overtime?
Hint: Tossing two different coins can be used to represent making or missing a shot. What are the outcomes when
you toss two coins? How can you use this to model the free throws?
you toss two coins? How can you use this to model the free throws?
Week 18 Challenge Question
Question: Every year, Arctic terns fly from the Arctic to the Antarctic and back, a distance of about 9000 miles each way. Suppose the birds fly at an average speed of 25 miles per hour for 12 hours a day. How many days of flying would be necessary to make the roundtrip?
Hint: How many miles would a tern fly in an average day?
Week 17 Challenge Question
Question: Is a discount of 30% off the original price, followed by a discount of 50% off the sale price, the same as a discount of 80% from the original price?
Hint: What would a $100 item cost after these discounts?
Week 16 Challenge Question
Question: Imagine that you bought a Beanie Baby™ for $6, sold it for $7, bought it back for $8, then sold it for $9. How much profit did you make?
Hint: Would it change your calculations if the second Beanie Baby™ were different than the first?
Week 15 Challenge Question
Question: Are any of the “National Champion” trees in the table below wide enough for a car to drive through?
| Tree | Girth (ft) | Height (ft) | Location |
| American Beech | 279 | 115 | Harwood, MD |
| Black Willow | 400 | 76 | Grand Traverse Co., MI |
| Coast Douglas Fir | 438 | 329 | Coos County, OR |
| Coast Redwood | 867 | 313 | Prairie Creek Redwoods State Park, CA |
| Giant Sequoia | 998 | 275 | Sequoia National Park, CA |
| Loblolly Pine | 188 | 148 | Warren, AR |
| Pinyon Pine | 213 | 69 | Cuba, NM |
| Sugar Maple | 274 | 65 | Kitzmiller, MD |
| Sugar Pine | 442 | 232 | Dorrington, CA |
| White Oak | 382 | 96 | Wye Mills State Park, MD |
Hint: The distance around a tree is its girth. The distance around a circle is its circumference. The “width” of a circle is its diameter. Finding the circumference of a circle involves the number π, about 3.14. The circumference of a circle is π times the diameter.
Week 14 Challenge Question
Question: In May 1999, two National League baseball players, Joe McEwing of the St. Louis Cardinals and Mike Lieberthal of the Philadelphia Phillies, each had identical batting averages (45 hits in 132 at bats). Suppose McEwing then batted .800 (4 hits in 5 at bats), and Lieberthal was perfect (3 hits in 3 at bats). Which player now has the higher batting average?
Hint: Batting Average = Number of Hits / Number of At Bats
Week 13 Challenge Question
Question: Is there a Friday the 13th every year?
Hint: If January 1 were on Monday, on what day of the week would January 13 fall? What about February 1 and February 13? Other months?
Week 12 Challenge Question
Question: Suppose you love chocolate. The tops of two differently shaped cookies are covered with the same thickness of chocolate. If you wanted to choose the cookie with more chocolate, which one would you pick?
Hint: Think about how to measure the area of the top of each cookie.
Week 11 Challenge Question
Question: Mark McGwire became baseball’s home run king in 1998 with 70 home runs. His 70th home run ball sold for slightly over $3 million in 1999. Babe Ruth, an earlier home-run king, hit 60 in 1927. His home-run ball was donated to the Hall of Fame. Suppose that Ruth’s ball was valued at $3000 in 1927 and, like many good investments, doubled its value every seven years .Would you rather have the value of Ruth’s ball or McGwire’s?
Hint: How many times would you need to double the value of Ruth’s ball to reach the value of McGwire’s?
Week 10 Challenge Question
Question: Spread out your fingers and look at your hand. Do people with big hands have larger angles between their fingers? Estimate the measures of the angles between your fingers when you spread out your hand.
Hint: When you hold your hand so that your thumb and index finger form an “L,” the angle formed measures about 90°.
Week 9 Challenge Question
Question: How much of yourself can you see in a small mirror?
Hint: Begin by measuring the height of the mirror and the amount of yourself that you can see.
Week 8 Challenge Question
Question: Suppose you found an old roll of 15¢ stamps. Can you use a combination of 33¢ stamps and 15¢ stamps to mail a package for exactly $1.77?
Hint: Use as many 33¢ stamps as you can so that the remainder can be made with 15¢ stamps.
This problem involves using combinations of numbers to make other numbers. Similar processes are used to develop codes to maintain security in banking and computer access.
Week 7 Challenge Question
Question: Would you rather work seven days at $20 per day or be paid $2 for the first day and have your salary double every day for a week?
Number patterns can change at very different rates. Understanding rates of change is important
in banking, biology, and economics.
in banking, biology, and economics.
Week 6 Challenge Question
Question: By walking on the sidewalk, how many different ways are there to go from home to the video store? No backtracking allowed.
Hint: Try fewer blocks to start. Counting is an important mathematical skill. Delivery companies and airlines count the number of travel routes to get from one place to another.
Week 5 Challenge Question
Question: What letters, when written in lowercase, can be read the same upside down as right side up?
Hint: Write out each lowercase letter and look at it in different ways. Symmetry is a basic geometric concept. Understanding how one part of an object mirrors the rest is important in art, design, medicine and other fields.
Week 4 Challenge Question
Question: Why aren't manhole covers square?
Hint: Investigate different shaped covers to see if they can fall through their corresponding holes. The shapes of many objects relate directly to their uses. Tools are designed with shapes that are easy to hold, furniture is designed with shapes that are comfortable, and race cars are designed to reduce wind resistance.
Week 3 Challenge Question
Question: If you like popcorn, which one would you buy? Take two identical sheets of paper. Roll one sheet into a short cylinder and the other into a tall cylinder. Set them both on a flat surface. Does one hold more than the other?
Hint: Place the taller cylinder inside the shorter one. Fill the taller one with dry cereal, rice, or popcorn; then remove it from the shorter cylinder. Which holds more?
Week 2 Challenge Question
Question: If you started counting your heartbeats at midnight on January 1, 2000, when would you count the millionth beat? What about the billionth beat?
Hint: Estimate your heart rate in beats per minute, per hour, and per day.
Week 1 Challenge Question
Question: How long do you think you would have to stand in line if you hold number 300?
Hint: Estimate the amount of time it would take for one person to buy a ticket. Use this estimate to find the amount of time you will have to wait in line.
Solutions to Each Week's Problem: