- Crossed Ladders Problem
The Crossed Ladders Problem is a
puzzle of unknown origin that has appeared in various publications and regularly reappears in Web pages andUsenet discussions.The Problem: Two ladders of respective length a = 40 and b = 30 feet lie oppositely across an alley, as shown. The ladders cross at a height of h = 12 feet above the alley floor. What is the width of the alley?
HIstory and Comment: Martin Gardner presents and discusses the problem in his book of mathematical puzzles published in 1979 and cites references to it as early as 1895. The Crossed Ladders Problem may appear in various forms, with variations in name, using various lengths and heights, or requesting unusual solutions such as cases where all values are integers. Its charm has been attributed to a seeming simplicity which can quickly devolve into an "algebraic mess" [characterization attributed by Gardner to D. F. Church] .
Puzzle Enthusiasts are warned that an outline of the solution and discussion is presented here, that several of puzzle's more "charming" traps are therein avoided, and the enthusiast may wish to consider whether to continue reading beyond this comment. It may be enough to know whether a
closed form solution exists. It does, although it is unlikely to be attained without the aid of a sixteenth-century mathematical adept, which is further hint. Discussion of the solution follows.Solution Summary: The problem may be reduced to the quartic equation "x 3(x - c) - 1 = 0", which can be solved by approximation methods, as suggested by Gardner, or the quartic may be solved in
closed form by Ferrari's method. Once "x" is obtained, the width of the alley is readily calculated. A derivation of the quartic is outlined below. Note the potentially confusing fact that the requested unknown "w", does not even appear!Setting up the Quartic Equation.
(Eq 1: 3 sneaky steps using similar triangles)
(Eq 2: Three easy steps, using the
Pythagorean theorem )**(Eq 3: Square (Eq 2) and combine with (Eq 1))
(same as Eq3)
Solve the above fourth power equation for x using your method of choice. Calculate the width of the alley using x.
References
* Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 62-64, 1979.
External links
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* [http://demonstrations.wolfram.com/CrossedLaddersTheorem/ Crossed Ladders Theorem] by Jay Warendorff,The Wolfram Demonstrations Project .
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