# A belt, a stone and a wooden stick

We're on the middle of the most devastating war the Earth has gone through all its history. You and your brother in arms are in a plane that has been taken down while crossing the enemies lines. You are okay, but your teammate has a head contussion and must be immediately brought back to your base.

Luckily, you are such a great pilot that the aircraft has nearly no damage and can fly again, but one of the two fuel tanks was lost due to the accident. Furthermore, the on-board computer has stopped working so it can't help you. You want to know if you can get your friend back to the base with the fuel you have. In the middle of your calculations, you find that you can't remember any of the digits of $\pi$, which you need to calculate the distance from where you are to your base (taking the Earth as a perfect Sphere) in order to estimate the fuel needed. There is nothing inside the plane which can help you with the calculation, so you have to stick to the next items.

You can ONLY use to calculate it:

• Your belt, which has a length of 1 metre.
• Some rocks you found near the plane.
• A wooden stick. You can use it to draw or anotate numbers on the ground.

Luckily, you can come up with a way to get a pretty good approximation of $\pi$ and be able to know if you can save your friend's live. Each second counts...

• Just to be clear, the puzzle is to approximate $\pi$ using only that equipment? How many digits is enough? – Moyli Oct 13 '14 at 7:48
• Are the rocks of equal size? – d'alar'cop Oct 13 '14 at 7:49
• The rocks are of equal size and 3 digits is enough. – Ioannes Oct 13 '14 at 7:53
• Isn't pi just equal to 4? that is easy enough to remember. – kaine Oct 13 '14 at 14:31
• Are you sure "you are okay" if you can't remember any digits of pi? – ThePopMachine Oct 13 '14 at 16:16

1. Lay your belt out flat and straight on the ground. It will act as the diameter. Count how many rocks you can lay across your belt. Call the number of rocks $x$.

2. Make a circle out of rocks around the belt where the belt is the diameter. Pivot the belt around if this helps to make as close-to-perfect a circle as you can. Count the number of rocks required to make this circle. Call the number of rocks $y$.

3. The remaining task is to find the ratio between $x$ and $y$. Perform long division with $y/x$. Using the stick on the sand to act as pen and paper. Calculate to $3$ digits.

PS: Since you only need 3 digits, you could just use $22/7$ or $355/113$ and skip steps 1 and 2. However, if such tricks were allowed, then surely one would know $\pi$ to the first 3 digits regardless and the puzzle would be pointless.

If you need to make navigation calculations you can probably use the handy fact that 1 angle minute (0°1') is exactly 1 nautical mile or 1852 km. If you know your coordinates and the destination coordinates, you should get a good estimate of how far to fly.

But I'm sure you won't leave before you figured out pi. For that, make a square of 200x200 stones. With the belt, measure the side S, and draw an arc of radius S from one of the corners. This gives a 1/4 circle in a square. The number of stones in the 1/4 circles should be 31416. Divide by 10000. (A few stones will be exactly on the circle. Count 1/2 for these).

1- Draw a $2\times 2$ square and inside it draw a circumference that is tangent to the sides of the square ($r=1$)

2-Start throwing stones to the square from the same position and keep a count on how many have you thrown ($N$) and how many landed inside the circumference ($n$). We assume that all of them land inside the square. If not, do not count that throw.

3- $\frac{n}{N}=\frac{A_{circumference}}{A_{square}}=\frac{\pi}{4}$

4-Approximate $\pi=\frac{4n}{N}$

The more stones you throw the more accurate the result is.

However, the method explained in the accepted solution optimizes the time spent in the calculation so it is a better solution.

Matematical idea:

We assume (and try to guarantee) that everywhere in the square has the same probability of being the place where a stone lands.

If this is true, then the probability of a stone landing inside the circle equals the ratio $\frac{A_c}{A_s}$ which is the fraction of area we count as a valid result divided by the total area. This seems intuitive as the bigger the square is in relation to the circle, the less likely a stone will land in it. And viceversa (Nearly equal areas implies nearly a 100% probability of landing inside the circle.)

Now we just have to put this reasoning into practise. By throwing $N$ stones and keeping a count of the ones that fall inside the circle ($n$) we'll get an empirical probability of a stone landing inside the circle ($\frac{n}{N}$) As the theoretical result should be the same as the empirical we can work with the equation in step 3 and get an approximation of $\pi$.

• That's a pretty cool idea. But could you expand out some of the mathematical reasoning? – d'alar'cop Oct 13 '14 at 12:05
• The "mathematical reasoning" is more of a requirement: You have to assume that stone can land everywhere inside the square with equal probability; while the requirement stated above, to throw all stones "from the same position", isn't technically required. In fact, throwing from different angles/positions could possibly help to equal out a bias towards the center. – Alexander Oct 13 '14 at 12:55
• I agree that throwing from the same position may not be the best analogy for equal probability... – Ioannes Oct 13 '14 at 14:01
• If my calculations are correct, you'd need to throw at least 4000 stones to get somewhere in the neighborhood of 3 decimal accuracy. If you manage to throw and record a stone every 5 seconds it would still take you about 5.5 hours to finish the task. – Moyli Oct 13 '14 at 20:01

Step 1. Draw a circle with the belt keeping the buckle as center.

Step 2. Lay stones over the belt.

Step 3. Lay stones along the circumference of the circle drawn with the belt.

Step 4. Count the number of stones over the belt, Say x

Step 5. Count the number of stones along the circumference, Say y

Step 6. Now calculate π = y / (2 * x) with the help of the stick, scribbling on the ground.

By this methos he can find out the value of π using his belt, stones and a stick.

First, you make a perfect circle with your belt (or as approximate as you can).

Then you mark the center of the belt and draw a straight line across the center.

Now count how many times the belt is bigger than the line (should be 3).