Lesson: Buffon's Needle Surprise


One of the reasons probability is a favorite attraction is that it can lead to many surprises along with some useful math done in interesting ways. The challenge in this activity is to see if students can make a prediction about the likelihood of toothpicks landing on lines when dropped randomly using experimental data. Target grades: 8-10.

  • Make a set of shoe box tops toothpick tossing “arenas”.
  • Draw equidistant, parallel lines inside the shoebox top or on a piece of 8.5 x 11 sheet of paper. The distance between the lines should be the same as the length of the toothpicks. Make one of these for each of your groups.
Note: If you don't have the boxes, you can also demonstrate the experiment by dropping the toothpicks on a tiled floor. Here I dropped a bunch on a hardwood floor. (See photo below.) I highlighted the toothpicks and lines on the floor. An adjustment will need to be made since the toothpicks will not be the same as the perpendicular distance between lines.

Setting the Stage

  • Have your students sit in groups or teams. You are facing them standing next to your one computer station.
  • Tell the students that they will be answering a question that was originally posed by a man named Georges-Louis Leclerc, Comte de Buffon about 250 years ago. But first...
  • Show this video (1:20). Ask the students what they notice and have them take 2 minutes to share with their neighbor(s). Ask volunteers to share what they noticed with the class. Repeat with what they wonder about. (If you are new to this "notice/wonder" strategy watch this video.)*
  • Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?
  • Take guesses from your class. Then ask them to explain their guess.
  • Ask them: How might we proceed to find out which student or group made the best guesses? Let’s do the experiment to help us find out.
Doing the Activity
  • Show them an "arena" for the experiment.
  • Hand out the activity sheet. Make sure the students understand the instructions.
  • Each group will drop a total of 100 toothpicks 10 at a time on their parallel lined arena.
  • Students toss the toothpicks and record them on the student sheet. When finished each group will share their results via a spreadsheet projected onto the white board by the teacher.
  • Students help the teacher to complete the summary spreadsheet. 
  • After finishing, ask the students how they might come up with even a better or best "guess"? (Theoretical probability is 2/Pi** or approximately .6366.) Who had the best estimate in the class? Compare the group results with the individual group guesses to determine who had the closest estimate. (That is, the group that came closest to .6366)
Debriefing the activity and what about the surprise?
  • Go to this website again.
  • Note the number that is constantly changing. Does it seem to be getting closer to a well known constant? It's Pi - approximately 3.14159.
  • How did Pi get into the act? Double the number of your individual throws and divide it by the number of crosses. What do you get? Check with other groups. What can you conclude? (Your answer is an approximation for Pi.)
  • Why does this happen? (See George Reese's explanation.)
Additional Resources
  • Here's alternative lesson plan.
  • Here's a cool article about Buffon's experiment.
  • Watch this simulation. 1:17 min video.
  • Another good example with some more advanced techniques.
  • Pi toss demonstration.
  • Estimating Pi 2:07 min video. Teacher TV. Buffon's needle experiment done using french bread and taped lines!
  • Imagine throwing a green dart onto an xy plane (dartboard) when a toothpick crosses a line and a red dart when the toothpick lands between the lines. After over 56,000 throws here's what the dartboard looks like.
  • To see the actual simulation done in Scratch go here.
  • In the simulation above, the number of throws was 56926 (35703 + 21223). So if we double the number of throws (113825) and divide by the number of crosses (green dart hits: 35703) you get 3.18... which is a reasonably close approximation to π.

*You can stop auto play in YouTube by switching to off the auto play switch which is in the upper right hand corner of YouTube.

**This formula works when the toothpick length is the same as the perpendicular distance between the lines.

Source: Original Version 1.0 developed at CIESE-
Center for Innovation in Engineering & Science Education (2007)
Revised 3.14.20 currently under construction