FOLDING RULERS INSIDE TRIANGLES

An l-ruler is a chain of n links, each of length l, The links, which a re allowed to cross, are modeled by line segments whose endpoints act as joints. A given configuration of an l-ruler is said to fold if it c an be moved to a configuration in which all its links coincide. We sho w that l-rulers confined inside an equilateral triangle of side 1 exhi bit the following surprising alternation property: there are three val ues x(1) approximate to 0.483, x(2) = 0.5, and x(3) approximate to 0.8 66 such that all configurations of n-link l-rulers fold if l is an ele ment of [0, x(1)] or l is an element of (x(2), x(3)], but, for any 1 i s an element of (x(1), x(2)] and any l is an element of (x(3), 1], the re are configurations of l-rulers that cannot fold. In the folding cas es, linear-time algorithms are given that achieve the folding. Also, a general proof technique is given that can show that certain configura tions-in the nonfolding cases-cannot fold.