The formula for the “Moving Stick Problem”

Overview

blackpenredpen derives the minimum length of a horizontal stick that can turn through a right-angle hallway with widths A and B. Minimizing L(θ) = A csc θ + B sec θ gives tan θ = (A/B)^(1/3) and the limiting length L = (A^(2/3) + B^(2/3))^(3/2); the closing question asks how the problem changes if the stick can be lifted under a ceiling of height C.

Key takeaways

For hallway widths A and B, a horizontal stick of length greater than (A^(2/3) + B^(2/3))^(3/2) cannot make the right-angle turn.
The stick length at angle θ is the sum of two geometric contributions: A csc θ along one corridor and B sec θ along the other.
The optimal orientation is determined by tan θ = (A/B)^(1/3), balancing the two corridor constraints.
A cube-root right triangle converts the optimal-angle condition into the compact minimum-length formula.
The derived formula assumes the stick remains horizontal; allowing it to lift beneath a ceiling of height C introduces a new three-dimensional constraint.

Chapters

0:00 Modeling the Horizontal Stick in a Hallway with Widths A and B

The hallway has perpendicular corridor widths A and B, and the stick stays horizontal while turning.
The limiting stick touches the outer wall edges at the corner; finding the longest stick that can pass reduces to finding this minimum limiting length.
At orientation angle θ, the stick divides into segments L1 = A csc θ and L2 = B sec θ.

3:00 Minimizing L(θ) to Find the Critical Turning Angle

The total limiting length is L(θ) = A csc θ + B sec θ, with θ between 0 and π/2.
Differentiating and setting the numerator equal to zero gives B sin³ θ = A cos³ θ.
Therefore tan³ θ = A/B, so the unique critical angle satisfies tan θ = (A/B)^(1/3).

7:20 Deriving the Optimal Length and Extending the Problem to a Ceiling

A right triangle with opposite side A^(1/3) and adjacent side B^(1/3) gives hypotenuse √(A^(2/3) + B^(2/3)).
Substituting the triangle ratios into A csc θ + B sec θ yields the minimum limiting length (A^(2/3) + B^(2/3))^(3/2).
blackpenredpen ends by asking how the result changes if the stick can be lifted and a ceiling of height C restricts its motion; no solution is provided.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, blackpenredpen.