You may also like

problem icon

Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

problem icon

Magic Caterpillars

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

problem icon

Plum Tree

Label this plum tree graph to make it totally magic!

Cube Net

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Congratulations Andrei, School No. 205, Bucharest, Romania on another excellent solution.

1. I observed that a cube could be represented by the diagram below, that keeps all edges and vertices (the lengths are not important). schlaefli diagram for cube
From the beginning I observe that one could start from any vertex in a cube and there are $8$ vertices.

From any vertex there are three possible routes. I shall consider the routes starting from vertex $A$.

Below are written all the possible combinations I found starting with the edge $AD$:

Path Circuit
A-D-C-B-F-E-H-G No
A-D-C-B-F-G-H-E-A Yes
A-D-C-G-H-E-F-B-A Yes
A-D-C-G-F - impossible
A-D-H-E-F-B-C-G No
A-D-H-E-F-G-C-B-A Yes
A-D-H-G-C-B-F-E-A Yes
A-D-H-G-F-E - impossible

I observe that there are $6$ possible paths, $4$ of which are Hamiltonian Circuits.

I have to multiply the number of solutions I obtained. So the total number of solutions is given by $8 \times3 \times6$ that is $144$, and $96$ are Hamiltonian Circuits.

Because all vertices of the cube are indistinguishable, there are $18$ solutions, and $12$ Hamiltonian Circuits.

2. Set $\{a, b, c\}$ has the following subsets: $\{a, b, c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a\}, \{b\}, \{c\}, \phi $. I observe that they could be arranged so that one subset is connected with $3$ other subsets that differ from the first by only one element, deleted or inserted. These subsets can be positioned on the vertices of a cube.
equivalent orthographic view of cuberelabelled schlaefli diagram
I have verified that each subset is connected with $3$ other subsets, forming a diagram as found before, or, more intuitively, a cube. So, the problem is reduced to the first problem.

The number of sequences is $144$ sequences, or $18$ if the first element of the sequence does not matter.