Why Colliding Blocks Produce Digits of π

Two colliding blocks can produce counts that begin with the digits of π. This lesson revisits the idealized physics, turns it into a geometry problem, and explains a subtle qualification about claiming the pattern for arbitrarily many digits. A later video explores an analogy with Grover’s quantum search algorithm.

Place a small stationary block between a wall and a larger block moving toward it on a frictionless surface. Count every block–block and block–wall collision. Equal masses give 3 collisions; a 100:1 mass ratio gives 31; 10,000:1 gives 314; and 1,000,000:1 gives 3,141.

The model assumes perfectly elastic collisions and ignores friction. The sounds in the animation are artistic: perfectly elastic collisions would not lose energy as sound. At very large mass ratios the small block would reach speeds where relativistic and other real-world effects matter, so the calculation is an ideal classical puzzle.

A problem-solving strategy is to list relevant laws. The total kinetic energy E = ½m₁v₁² + ½m₂v₂² remains constant. The blocks’ total momentum m₁v₁ + m₂v₂ is conserved during a collision between them, but changes when the small block hits the wall because momentum is transferred to the wall.

Represent the two velocities as a point in a plane. As collisions change v₁ and v₂, the point moves. In these coordinates, constant kinetic energy gives an ellipse; initial speed changes the scale and timing but not the collision count.

Rescale the coordinates to x = √m₁ v₁ and y = √m₂ v₂. Now energy conservation becomes x² + y² = 2E: a circle. This change of coordinates makes the symmetry behind π visible.

In the rescaled plane, constant momentum is a straight line √m₁x + √m₂y = P with slope −√(m₁/m₂). A block–block collision moves the state between the two intersections of that line and the energy circle. A wall collision flips the sign of y and changes the blocks’ total momentum.

The sequence ends when both blocks move away from the wall and the smaller one cannot catch the larger one. In state space this is a wedge between the positive x-axis and the line corresponding to equal velocities. The physical problem is now a sequence of straight-line moves around a circle.

The inscribed-angle theorem shows that successive relevant arcs span equal angles of 2θ, where θ is the angle between the vertical direction and a diagonal collision line. Counting collisions becomes a question of how many such steps fit before reaching the end region.

Equivalently, the count is the largest integer N for which Nθ < π. The strict inequality matters if π/θ happens to be an integer: in general N = ⌈π/θ⌉ − 1. This is an exact geometric count for the stated ideal setup.

The slope gives tan θ = √(m₂/m₁), so θ = arctan √(m₂/m₁). If the mass ratio is 100:1, θ = arctan(0.1), not exactly 0.1. For small angles arctan x is close to x, explaining why the count resembles digits of π.

On a unit circle, a small angle in radians is close to its arc length and to the vertical displacement; the horizontal coordinate is close to one. This geometric picture explains the small-angle approximation used in the argument.

Close is not automatically enough to prove every digit. A tiny approximation error could change the integer count if a multiple lands sufficiently close to π. The video notes a possible off-by-one issue related to unusually long runs of nines in π’s decimal expansion; it says a general proof for all powers of 100 remains open.

The collision count itself has an exact formula, so the physical model is not mysterious or unsolved. The unresolved issue described in the video is the universal digit-matching claim. Other number bases have analogous mass-ratio constructions, such as powers of four for binary digits.

Idealization has two uses. A simple solution can be refined to incorporate real effects, such as energy loss; it can also expose connections hidden by physical detail. The presenter points to Gregory Galperin’s mirror analogy and a separate forthcoming explanation of the Grover algorithm connection.

Distilling a problem to its essential structure can reveal a clearer version of the same difficulty in another setting. Those connections are a source of mathematical progress, even when the original physical experiment is impractical at extreme scales.