Possible Increasing Paths in Cubes
Given a cube, a valid label is one where each edge is labeled a distinct number from 1 to 12. An increasing path is a path formed by the edges labeled \(i, j\) and \(k\) such that \(i < j < k\) and the edges \(i, j\) and \(k\) form a continuous curve (i.e. edge \(j\) has a common vertex with edges \(i\) and \(k\), and these 3 edges do not share a common vertex). Over all possible valid labels, what is the minimum number of increasing paths?