What it shows
Rice grains scattered evenly over a square land inside the circle drawn within it with probability equal to the circle's share of the area, which is pi divided by 4. Counting grains inside and outside and multiplying the inside fraction by 4 estimates pi, a Monte Carlo method used in real computer simulations.
The science
A circle of radius r drawn inside a square of side 2r has area pi r squared, while the square has area 4 r squared, so the circle covers pi over 4 of the square, about 78.5 percent. If grains fall at random, the fraction landing inside the circle approaches this share, and 4 times the fraction approaches pi. The estimate improves slowly: to gain one more correct decimal place you need about 100 times more grains, which is why computers use millions of random points.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A square tray or a 30 cm square drawn on chart paper with a tape border
- A compass to draw the largest circle inside the square
- About 500 grains of rice or mustard seeds
- A tally sheet and a calculator
- A sieve or a sheet to scatter grains from a height
Safety
This project uses safe, everyday materials. Still, read the rules below with a parent before you start.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
7 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw a 30 cm square and the circle of radius 15 cm that just touches its sides. Make the lines thin and clear.
Hold a handful of about 100 grains high above the centre and let them fall, so they spread evenly. Grains that land outside the square are ignored.
Count the grains inside the circle and the total inside the square, counting grains on the circle line as inside.
Calculate 4 times inside divided by total and record it as your first estimate.
Repeat four more times, keeping a running total of inside and total counts, and recompute the estimate after each round.
Plot the estimate against the total number of grains and watch it settle near 3.14.
Explain in a note why a fair scatter matters: grains dropped from low height pile up in the middle and give too high a value.
What you should see
After about 500 grains the estimate lies between roughly 3.0 and 3.3, and the running estimate wanders less with every round, showing the method converging on pi.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why is the inside fraction close to pi over 4?
Because the circle covers pi r squared out of the square's 4 r squared, and random grains land in proportion to area.
2What is a Monte Carlo method?
A way of estimating a quantity by random sampling and counting, named after the casino.
3If 393 out of 500 grains fall inside, what is the estimate?
4 times 393 divided by 500, which is 3.144.
4How does this differ from Buffon's needle?
Buffon drops needles on lines and counts crossings; this drops points and counts those inside a circle. Both use random trials to estimate pi.
Take it further
Write a ten-line program that drops a million random points and compare its estimate with your rice result.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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