Problems to Ponder

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  • 1 Chesterland, Ohio
  • 2 Fiske Elementary School, Lexington, Massachusetts

Problems to Ponder provides 28 varied, classroom-ready mathematics problems that span grades PK-12, arranged in order of grade band. Links to the problem answers are available in this department.

1. Find the numbers that match.

2. Using the shapes, create a pattern. You can add color to the shapes.

3. Hypatia rolled three dice and counted all of the dots. She counted 10 dots. She did not roll any ones or sixes. What three combinations of numbers might she have rolled?

4. On Saturday, Brooks was at his home. Brooks walked 2 blocks north to Max's house. Then, Brooks walked 5 blocks west to Sophie's apartment. Then he walked back to Max's house and then back home.

If Brooks stayed on the sidewalk and did not take any shortcuts or detours, how many blocks did Brooks walk?

5. Using the numbers on the board as many times as you want, what sums can be made between 6 and 23? What sum can be made in exactly five ways? (For this problem, we will consider 5 + 2 + 5 = 12 and 5 + 5 + 2 = 12 to be the same.)

6. Two numbers added together sum to 20. One number is 10 greater than the other. What are the two numbers?

7. Maryam has 15 flowers. Some are red, some are yellow, and some are purple. If she has more yellow than red, and more purple than yellow, how many of each color might she have if the number of each color is odd?

For further thought: Could they all be even? Why or why not?

8. Using three different single-digit numbers, how can you make this statement true?

9. Continue this pattern in different ways.

Ant, Bear, Camel…

10. Ada picked three polygons out of a bag. She counted all the sides of the shapes. Ada counted 12 sides. What shapes might she have picked out of the bag?

11. Owen's birthday cake is a rectangle 12 inches long and 9 inches wide. It is frosted on the top and on all four sides. Owen cuts the cake into 3-inch squares. What is the number of pieces frosted on three sides and the number of pieces frosted on one side?

12. The sum of three prime numbers is a square number. What might the number be?

For further thought: Can you find any square numbers that are not the sum of three primes?

13. What is staying the same? What is changing? Sketch the next two cases. How many stars appear in case 11?

14. These triangles follow a rule.

What number replaces each variable?

15. A drawer contains 5 black, 5 red, 5 blue, and 5 white socks. If you remove socks at random from the drawer, what is the minimum number of socks you must remove to be certain you have a pair that match?

For further thought: How does this change if you start out with 10 of each color? How does this change if you add green socks, so that there are 10 socks of each of 5 colors?

16. The first two terms in a Fibonacci-like sequence are specified. Each subsequent term is the sum of the previous two terms. One example (the Fibonacci sequence) is 1, 1, 2, 3, 5, 8, 13, …. Come up with another Fibonacci-like sequence whose fifth term also has a value of 5.

For further thought: How many different Fibonacci-like sequences are there whose fifth term has a value of 5?

17. A rope encircles the equator of a spherical asteroid with a diameter of 5,000 meters. How long is the rope?

18. If I drive for 3 hours at 40 miles per hour, 2 hours at 50 mph, and 3 hours at 60 mph, what is my average speed?

19. Alice and Belle are born on the same date, but in different years. One year Alice is exactly 4 times as old as Belle. Exactly one year later she is 3 times as old. How old was Alice when Belle was born?

20. A circular pizza with a 16-inch diameter is cut into 8 equal slices. A square pizza is cut into 9 equal-size pieces. To the nearest inch, find the dimensions of the square pizza so that each of its 9 pieces has the same area as one slice of the round pizza.

For further thought: If a box is made to exactly fit the square pizza, will the circular pizza fit into the box?

21. Three merchants originally sell an item at the same price. Over the course of several weeks, Merchant A raises the sales price by 10 percent, raises it by another 10 percent, and then lowers it by 20 percent. Merchant B raises the price by 20 percent, lowers it by 10 percent, and then lowers it by another 10 percent. Merchant C raises the price by 20 percent, and then lowers it by 20 percent. Order the merchants' final prices from lowest to highest.

22. Jack walked up and down a hill. The hill's shape is a cone frustrum (a cone with the tip cut off) whose sides make a 30-degree angle with the ground, with a vertical height of 10 feet. Jill went around the hill, which has a circular base with a diameter of 50 feet. Who walked farther?

23. Newton is a dog who is attached to a 25-foot-long leash that is attached at point O as shown in the diagram below. He runs in a semicircular arc from point A to point B and back again to point A. Gauss is a dog who is unleashed. She runs in straight lines from point A to point C, point B, and back to point A. If both dogs run at a speed of 10 feet per second, which one returns to its starting point first?

24. Four road signs, as shown below, are exactly the same height. The YIELD sign is an equilateral triangle, the STOP sign is a regular octagon, the RR sign is a circle, and the CAUTION sign is a square. Order the following road signs from least area to most area.

25. The cost of a 30-second Super Bowl television advertisement was $3 million in 2011, and it was $4.5 million in 2015. Estimate the cost in 2017 if the costs increase linearly, and if the costs instead increase exponentially.

26. Start with a temperature measured in degrees Fahrenheit and then add 50°F to it. Convert both starting and ending temperatures to degrees Celsius. What temperature must you start with so that the second Celsius temperature is twice the first?

27. A meteorologist predicts a 40 percent chance of rain each day in the next week (7 days). If the meteorologist is perfectly accurate, what is the probability that there will be rain on at least one day during the week?

28. Planet X and Planet Y have circular orbits around their sun, with radii of 150,000,000 and 225,000,000 kilometers, and orbital periods of 350 and 643 days, respectively. Consider a time when both planets are at their closest distance to each other. How many days until they are again at their closest distance?

For further thought: How many days until Planets X and Y are at opposition (in a direct line, but on opposite sides of their sun)?

Answer key available at

Steve Ingrassia,, is a retired high school mathematics and computer science teacher in Chesterland, Ohio. He is interested in applying multiple techniques to solving cross-disciplinary, real-world problems.

Molly Rawding,, is a mathematics specialist/coach at Fiske Elementary School in Lexington, Massachusetts. She enjoys collaborating with teachers to develop students' mathematical understandings through visuals and puzzles.

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Supplementary Materials


Problems to Ponder provides 28 varied, classroom-ready mathematics problems that collectively span PK–12, arranged in order of grade level. Answers to the problems are available online.


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