MathDB

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 \equal{} B.

Problem Statement