# Calculating the digits of pi by colliding boxes

• I
• Frodo
In summary, the demonstration and the proof show that if you increase the right box by a certain amount, the next digit of pi will be generated.
Frodo
Gold Member
TL;DR Summary
Calculating the digits of pi by colliding boxes
I came across this and it is rather fun!

Assume there is a floor and a wall. There is a 1 lkg box on the left and a box to its right as shown in the diagram. Assume there is no friction and that no energy is lost during any collision.

Set the right box to 1 kg and cause it to move to the left until it collides with the 1 kg box. It stops and the left box is pushed to the wall where it bounces off, collides with the right box and stops. The right box travels off to the right. Count the total number of collisions: there are 3.

Repeat by setting the right box to 100 kg; 10,000 kg; 1,000,000 kg ..., etc. We have:

Set the right box to 1 kg and count the collisions: there are 3
Set the right box to 100 kg and count the collisions: there are 31
Set the right box to 10,000 kg and count the collisions: there are 314
Set the right box to 1,000,000 kg and count the collisions: there are 3141

Each time you increase the right box by 100x, you generate an additional digit of pi.

The demonstration and the proof are given at The most unexpected answer to a counting puzzle and the referenced explanation videos.

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In a somewhat similar vein, David Bailey and colleagues discovered in 1995 a formula for calculating the Nth digit of pi without calculating the preceding digits. See Finding the N-th digit of Pi which says

By "somewhat similar" is mean that the method similarly allows one to ignore later terms so as to force the integer answer as does the colliding boxes problem.

Frodo said:
Summary:: Calculating the digits of pi by colliding boxes
Astounding! I never would have believed it. Thanks for posting.

EDIT: for other readers of this thread, the real fun is in the follow-on videos which dig into the math:

Part 2 (15 minutes)
Part 3 (14+ minutes)

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## 1. How does colliding boxes help in calculating the digits of pi?

The idea behind using colliding boxes to calculate pi is based on the concept of Monte Carlo simulation. By repeatedly colliding boxes within a given space, we can estimate the value of pi by calculating the ratio of the boxes that intersect with the walls of the space to the total number of boxes.

## 2. What is the mathematical formula used in this method?

The formula used in this method is pi = (2 * N) / T, where N is the number of boxes that intersect with the walls and T is the total number of boxes.

## 3. How accurate is this method in calculating pi?

The accuracy of this method depends on the number of boxes used in the simulation. The more boxes we use, the closer our estimate will be to the actual value of pi. With a large number of boxes, we can achieve a high level of accuracy.

## 4. Can this method be used to calculate other mathematical constants?

Yes, this method can be applied to calculate other mathematical constants as well. It is a general method for estimating the value of a constant by using Monte Carlo simulation.

## 5. What are some potential applications of this method?

This method can be used in various fields such as physics, engineering, and finance to estimate the value of pi. It can also be used to test the accuracy of other methods for calculating pi or to demonstrate the concept of Monte Carlo simulation in educational settings.

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