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 and Usenet discussions.

The problem

Two ladders of lengths 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?

Crossed ladders. h is half the harmonic mean of A and B

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

$A B = h (A + B) \,$         (Eq 1: Three sneaky steps using similar triangles)

$a^2 - b^2 = (B + A) (B - A) \qquad{}$         (Eq 2: Three easy steps, using the Pythagorean theorem)

$(a^2 - b^2)^2 = (A+B)^3(A+B-4h) \,$         **(Eq 3: Square (Eq 2) and combine with (Eq 1))

Let

$x = \frac{A + B}{\sqrt{a^2 - b^2}}, \,$

$c = \frac{4 h}{\sqrt{a^2-b^2}}. \,$

Then

$x^3(x-c)-1 = 0.\,$         (same as Eq 3)

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.

A quartic equation has four solutions, and only one solution for this equation matches the problem as presented. Another solution is for a case where one ladder (and wall) is below ground level and the other above ground level. In this case the ladders do not actually cross, but the intersection of their extensions do so at the specified height. The other two solutions are a pair of conjugate complex numbers where presumably the imaginary dimension is normal to the plane of the diagram. The equation does not have the ladder lengths explicitly defined, only the difference of their squares, so one could take the length as any value that makes them cross, and the wall spacing would be defined as between where the ladders intersect the walls. Surprisingly as the wall spacing approaches zero, the height of the crossing approaches

a.b/(a+b)   

As the solutions to the equation involve square roots, negative roots are equally valid so both ladders and walls can be below ground level and with them in opposing sense, they can be interchanged.

External links

• Weisstein, Eric W., "Crossed Ladders Problem" from MathWorld.
• Weisstein, Eric W., "Crossed Ladders Theorem" from MathWorld.
• Crossed Ladders Theorem by Jay Warendorff, the Wolfram Demonstrations Project.
• Solving the crossing ladders puzzle (with Python, GNU GSL, Octave, Maxima and Sage).

When the crossing point is not between the walls, (when one ladder is above ground and the other below)) then the maximum height of the crossing is given by a./(a - b)