Introduction: Unlocking Pi With Matchsticks: Discover Hidden Patterns in the Randomnness of Universe

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About: - Teacher, Inventor. ~There are many problems in the world to solve~ ~Innovation gives fulfilment to me~

I'll keep it short at first so readers know what they are going to get:

  1. Drop matchsticks onto a sheet of paper that has some spaced lines.
  2. Count how many matchsticks cross lines.
  3. Calculate a value using the formula: 2×(Total matchsticks/Matchsticks that cross a line)
  4. This value approximates Pi, and the more you repeat the experiment and average the results, the closer it will get to Pi, according to statistical theory.

How to do this, Why this works, and how it can raise curiosity in maths for young students, all are described below.

This is to showcase a surprizing "orderly random" phenomenon. We are making visible the hidden patterns that govern even the most random occurrences.

Supplies

  1. Two paper-sheets of A4 size or similar . This needs to be large enough so that when you drop the matchsticks, the matchsticks do not cross the boundary of the papers.
  2. Rulers to measure and draw the lines
  3. Matchsticks (80 to 100 numbers to get accuracy around +-0.5 . More numbers=more accurate)
  4. Pen/ pencil to draw lines.
  5. Transparent tape for sticking papers to a flat surface.

Step 1: Draw Line

  1. Measure the length of a matchstick. For my case, it was 4 cm.
  2. You have to draw parallel lines on the sheet of paper at 4 cm distances starting from one edge.
  3. Mark points at 4 cm distances on each end of the paper so the lines are exactly parallel when connected.
  4. Then draw line by using a ruler or the straight edge of the notebook as shown.

Step 2: Set Paper on Flat Surface

  1. Tape the paper to ground/ or table.
  2. you may use 4 or more papers to get results more accurately.

Step 3: Drop and Count


  • Drop the matchsticks from few inches above. Make sure of the following:
  • Try to minimize overlapping of the sticks so counting becomes manageable.
  • Dont throw/roll along paper as it may roll the sticks and give biased results due to alignment.
  • Collect the matchsticks as: crossing the lines at one side vs Not crossing on the other side.
  • Count the matchsticks. In the video, Crossing=55 nos, Not crossing= 33 nos
  • Value of Pi calculated= 2*(total matchsticks)/(Crossing numbers)= 2*88/55=3.2.
  • Actual Value=3.14. (Error= 1.9%)
  • You can do multiple tests, I did and found the values could be varying within +-0.5 (from 2.64 to 3.64) .The more you do the tests, closer your average gets to 3.14.

Step 4: Theory

Proving the theory could be presented as a challange for a Grade 11 on-wards student or even a Graduate for fun.

An easy version with Geometry+integration is shown below. See images for better understanding. (kindly feel free to ask me any questions good or bad):

  • Say the matchsticks are of unit length 1. And a matchstick is at an angle "a" from parallel lines.
  • Then we plot the distance in Y axis from 0 to 0.5 and rotation at X axis from 0 to "pi".
  • The portion which shows overlap of matchstick with parallel lines, is shown in chequered area.
  • Find the integral of the area =1: see image (writing equation courtesy "mathcha.io online editor")
  • Integral of the rectangular area is 0.5Pi
  • Hence, the probability of overlap=1/(0.5*Pi) =2/Pi
  • From this equation, overlap/total=2/Pi.
  • So, Pi = 2*(total matchsticks)/(Overlap)

Step 5: What Importance Can This Simple Project Have?


Students often say about math, 'Math is hard because I don't understand the formulas,' and 'What's the point of memorizing this?'.. What they actually want to say that they are not interested to learn formulas . I think that a good teacher should firstly show how a lesson is needed, or used, in real life and why we should be interested. When we know why a formula important, it becomes interesting. When it becomes interesting, We give time to learn it. And when we learn it, it becomes easy. You cannot skip any step. The problem of our text-book based educational system is, we sometimes push students directly to memorize without telling its usefulness. Then it becomes hard just because it is new. And it becomes kind of frustrating to learn something we don't find useful. So try a different approach: get them interested through games and then show them how can they also find answers to nature's secrets. This will surely inspire many students in STEM (Science, technology,Engineering, Mathematics) field because we need good ideas to save our world in ever growing critical global problems.

This is why I particularly put emphasis on "Experiment and fun" based learning. I get good results in students. Please have a try if you happen to teach any kid and let me know your results!

Step 6: Reference Idea

The original idea of this work was formulated in 1777 and later published in Buffon's work, "Essai d'Arithmétique Morale." His goal may not have been to get value of Pi. However, the result of his probabilistic experiments gives results including Pi. Hence this idea can be utilized in the reverse to showcase the value of Pi as well.