How long a pole can you get through a dungeon corridor?
Obviously the correct answer is to make it sectional, with screw
or cup fittings to hold it together. But assume you can't do that.
The dungeon corridor is assumed to be W wide and H high. We also
assume a rigid pole of length L, of trivial diameter.
The first step is to find the maximum length in two dimensions. The
corner is a W × W square. We'll put one end of the pole at the outside
corner and lay the rest of it along wall A in position ①; then we'll
slide the front end along wall B while the back end slides along wall
A into the corner, though positions ②-⑥, ending up in ⑦. The
constraint is that the pole can't penetrate the inside corner C.
So let's put the outside corner at (0, 0) in our coordinate system.
The pole in its starting position runs from (0, 0) to (0, L), and at
the end from (L, 0) to (0, 0).
I'll call the non-zero coordinates X (the X coordinate of the front
end) and Y (the Y coordinate of the back end).
If the pole will jam at all, it will be jammed when it's at 45 degrees
relative to the corridor (position ④), so that's the position I'll
consider: if it is still free to move there, it'll be free to move
during the rest of the procedure. Since it's L long, its coordinates
must always be constrained by (X² + Y² = L²); at this half-way point,
X = Y, and therefore X = Y = L/√2.
At this moment, the location of the midpoint is half-way between (X,
0) and (0, Y), so we can halve these coordinate values again to find
(L/2√2), which is both the x and y coordinate. This number cannot
exceed W (the x and y coordinate of the corner C) in order for the
pole to fit. So L ≤ 2√2 × W. (In a ten-foot corridor, we can corner
with a pole of 28 feet 3 inches.)
This two-dimensional solution is also the plan view of the
three-dimensional solution, because we can put one end of the pole on
the floor while the other scrapes the ceiling: the maximum pole length
is now √(L² + H²) or √(8 × W² + H²) In the 10-foot corridor with
15-foot ceilings, that's now up to 32 feet.