ILL 1990 Problem 60

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $A$ be the beginning vertex and $B$ be the end vertex. Let there be $p \times q \times r$ cubes on the string $(p, q, r \geq 1).$

(a) Determine for which values of $p, q,$ and $r$ it is possible to build a block with dimensions $p, q,$ and $r.$ Give reasons for your answers. (b) The same question as (a) with the extra condition that $A = B.$