At-home STEM Activities: Buffon's Needle Problem
The Discovery Center’s mission is to inspire every generation to reach for the stars, through engaging, artful and entertaining activities that explore astronomy, aviation, earth and space science. So while were closed and while our visitors are finding the best ways to learn at a distance, we’ll be posting some activities here with the goal of helping to enrich kids’ education at home.
Pi (π) is a mathematical constant that has captured the interest of people for thousands of years. In a geometrical sense, pi is the ratio of a circle’s circumference (the distance around it) divided by its diameter (the distance across it). Using this definition, people have been approximating pi since the ancient Babylonians estimated pi to be about 3.125 circa 1900-1680 B.C.E.
Pi is an irrational number. An irrational number is a number that cannot be expressed as a fraction—that is, it can’t be written as one whole number over another. So even though pi is a ratio of two measurements of a circle, both the circumference and the diameter cannot be whole numbers for one circle. An irrational’s decimal also goes on forever, so the pi has an infinite decimal that has no pattern.
Math is the language scientists use to describe the world around us. In the equations scientists use, pi frequently comes up. Since pi is so closely related to circles, most things that occur cyclically, from the orbit of planets to the ebb and flow of tides, can be modeled using pi. In that way, pi can sometimes show up in unexpected places—try the activity below to one of these situations!
Buffon’s Needle Problem
In 18th century French gambling halls, people liked to play a game where they would drop a needle on a wooden floor, betting on whether or not the needle would cross the lines between the floorboards. Georges-Louis Leclerc, the Comte de Buffon, saw this game being played and asked himself, “What is the probability of the needle crossing the line?” So he went home, dropped a needle on his floorboards over and over, and, surprisingly, found that pi involved. Run the Comte de Buffon’s simulation, and see for yourself how pi comes up!
Materials:
2 pieces of paper (one at least 8.5”-by-11” and one of any size)
Marker or other writing utensil
Ruler
Toothpick
Instructions:
1. Measure the length of your toothpick. On your piece of paper (the one that’s at least 8.5”-by-11”), draw lines such that the distance between each line is the length of the toothpick.
2. Hold the toothpick several inches above your lined paper and drop the toothpick. Record if the toothpick crosses a line or not. If the toothpick rolls of the paper or you can’t tell if it’s crossing a line, just drop it again.
3. Repeat step 2 many times—the more times you drop the toothpick, the more accurate your results will be. We recommend at least 100 times.
4. Calculate the probability of the toothpick crossing a line by dividing the number of crosses by the total number of drops. We’ll give this value a name: P.
5. Now, divide 2 by π. Compare this value to P. Are they similar?
You can find how similar they are by calculating the percent difference. The percent difference is the difference between the two values (subtract the two values) divided by the average of the two values (add the values and divide by 2), expressed as a percentage.
Based on our simulation, the probability of the toothpick crossing a line is 2/π
Why do you think that is?
As we said above, pi is related to circles. When dropping your toothpick, did you notice any way circles could be involved?
The toothpick can land at any angle relative to the lines. If we combine all of those possible angles, we get a circle. Once we have a circle, pi’s involvement becomes clear.
To see why the probability is exactly 2/π requires some calculus The University of Illinois has a good explanation on their website, if you are interested in seeing that. They also have an online simulation, if you are missing any of the material for this activity.
Take this activity further:
Play around with scale: You can do Buffon’s Needle Problem with any size needle. Try running a simulation by taping masking tape lines to your floor a paper towel tube’s distance apart, or drawing chalk lines on your driveway a pool noodle’s length apart!
Change the distance between the lines: What do you think the probability of crossing a line is when the distance between the lines is greater than the length of the toothpick? When it’s less?
Try Buffon’s Circle Problem: Try swapping out the toothpick for a coin. What do you think the probability is of crossing a line now?
