This is my version of "the moving ladder problem": Return to my Puzzle pages
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© Copyright 2003, Jim Loy
This is my version of "the moving ladder problem":
We have two hallways, width 2 and 3, which meet in a corner. We want to know the length of the longest ladder (line segment actually) which can be carried around the corner (with the ladder carried with each end no higher than the other).
If the hallways have the same width a, then the maximum ladder is 2a sqrt(2), where sqrt(2) is the square root of 2. See Mathsoft - Moving Sofa Constant, where the hall widths are both 1. Also, if we are allowed to tilt the ladder, with one end higher than the other, that is the same problem with a shorter ladder (and probably a broken light bulb).
If the hallways are of different width, as I have stated the problem, then it becomes more complicated. Above right is a curve that I drew to find the minimum ladder length c, given hallway widths a and b. I drew a line segment AB representing the maximum ladder for any A. We find the minimum c with A positioned as shown. Longer ladders (greater c's) become stuck in the corner before they reach this position. Looking at that curve suggests that the solution is not easy. Measurement of a bigger diagram shows that the ladder has a length of 7.02.
Using this diagram, and similar triangles,
we get y = ab/x. Then using The Pythagorean
Theorem, we get c^2 = (a+x)^2 + (b+y)^2, where c^2 means c squared. We
eventually get a fourth degree equation which we have to minimize. We do this
by taking the derivative of the fourth degree equation (see
Derivatives), which probably gives us a third
degree equation. Setting the derivative to zero gives us a minimum. That may be
a difficult problem.
Using our widths of 2 and 3, there are several ways to estimate the result. I chose to write a computer program to evaluate the above equation for various values of x, and print out the value of c when it becomes a minimum. Then I get an answer of 7.0234824.
Someone sent me an algebraic solution to the above problem. But I lost it during a computer crash.
A similar problem is the moving sofa problem (see Mathsoft - Moving Sofa Constant). Also see The Crossed Ladder Problem and Another Ladder Problem.