Abstract
The so-called ladder problems have attracted the interest of mathematicians through centuries. For an overview and a historical bibliography, see [3] and [4]. One such problem is the crossed ladders problem (CLP), which can be formulated as follows: Two ladders of length a and b lean against two vertical walls. The ladders cross each other at a point with distance c above the floor (see Figure 1). Determine the distance x between the walls and the height above the floor y and z where the ladders touch the walls. The existence of integer solutions of this problem is well known (see [1], [6], [8] and [7]). However, the actual calculation of integer solutions using known methods is cumbersome. In this paper we introduce essentially simpler methods for calculating all possible integer solutions by means of four parameters. We also identify several interesting sub-classes of CLP-solutions described by fewer parameters or obtained recursively by using sequences of Pell type. A study of minimal integer values is also included. Minimal integer values have been discussed earlier in [6] and [7]. In this paper we formulate new methods for identifying such values and present values smaller than the ones formerly published. The paper is organized as follows. In Section 2 we give some remarks concerning Pythagorean triples and the trivial case a = b, Y = z.Parametric representations of general solutions are collected in Section 3. An alternative method for parametric representation of several interesting subclasses can be found in Section 4. In Section 5 and Section 6 we study minimal integer values, in particular the minimal value of x, which appears in a certain subclass of CLP-solutions. The connection between this subclass and recursive formulae of type Pell numbers is found in Section 7.Similarly, in Section 8 we describe some subclasses of CLP-solutions containing minimum values for c and z.