Demonstrating “number of collisions = pi” in a real-world experiment suspended apparatus minimizing friction reproduces 31 collisions (3.1) at a 1:100 mass ratio

A research group including Okayama University of Science (OUS) has successfully reproduced, in real-world experiments, the previously established theory that the number of collisions between two objects and a wall corresponds to the value of pi (π).

Using a suspended apparatus designed to minimize energy loss, the team repeatedly confirmed 31 collisions under a mass ratio of 1:100—corresponding to 3.1, the first decimal place of pi. Their findings were published in the European Journal of Physics.

Background
In the 1990s, it was theoretically demonstrated that the number of collisions between two objects and a wall corresponds to the digits of pi. For example:
･ With a mass ratio of 1:1, the total number of collisions is three (matching the integer part of π).
･ With a mass ratio of 1:100, the total is 31 (corresponding to 3.1).
･ With a mass ratio of 1:10,000, the total is 314 (corresponding to 3.14).
In real-world setups, the influence of friction and rotational motion made it difficult to reproduce the theoretical number of collisions.
In this study, the researchers suspended the objects in midair rather than placing them on a surface, thereby minimizing friction and enabling the theoretical number of collisions to be measured experimentally.

Key Experimental Points
Apparatus: The objects were suspended and allowed to oscillate, with collisions counted both between the wall and between the two objects.

Result: With a mass ratio of 1:100, the team repeatedly measured a total of 31 collisions (corresponding to 3.1), thereby confirming the theory through real-world experimentation.

Key Points
･ Successfully measured 31 collisions, corresponding to 3.1 in π in the experiment, corresponding to the first decimal place of pi (π).
･ Using an experimental setup designed to minimize energy loss, the research team demonstrated that the theoretically known relationship between the number of object collisions and pi can also be reproduced under real-world conditions.

Overview of the Findings
It has long been known in theory that the number of collisions between two objects and a wall corresponds to the digits of pi (π). However, for this relationship to hold, idealized conditions without friction or resistance are required, making it difficult to measure pi through this method in real-world environments.

In this study, the research team designed an apparatus that suspends the colliding objects in midair, thereby minimizing energy loss from friction. With this approach, the team successfully measured 31 collisions—corresponding to 3.1, the first decimal place of pi—and demonstrated that π can indeed be derived experimentally under real-world conditions.

Research Background
In the 1990s, it was theoretically demonstrated that the number of collisions between two objects and a wall corresponds to the digits of pi (π).
･ For example, with a mass ratio of 1:1, when one object P is pushed toward another stationary object Q, Q begins to move. Q collides with the wall, bounces back, and collides once more with P. No further collisions occur, so the total number of collisions is three—corresponding to the leading “3” in pi(3.14…).
･ With a mass ratio of 1:100, when the heavier object collides with the lighter object, a total of 31 collisions occur before all collisions end. This corresponds to 3.1, the first decimal digit of pi.
･ Similarly, with a mass ratio of 1:10000, the collisions total 314, corresponding to 3.14, the first two decimal digits of pi.
This curious relationship between collision counts and pi, which at first glance seems unrelated, has attracted attention and even been featured on platforms such as YouTube. However, while the theory became well known, reproducing it experimentally proved difficult. In real-world setups, frictional energy loss and the rotational motion of objects prevented the theoretical number of collisions from being achieved.

New Achievement
The research group successfully measured a total of 31 collisions—the theoretical value corresponding to 3.1—multiple times under a 1:100 mass ratio between two objects and a wall. This demonstrated that pi (3.1) can be
experimentally derived.

To determine pi from collision counts, an idealized environment without energy loss from friction or resistance
is required. In this study, instead of allowing the objects to move along the floor, the researchers suspended them in the air. This novel approach eliminated friction between the objects and the floor, making it possible to reproduce the theoretical value experimentally.
 

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