Challenge Problem. Assign distinct positive integers to the edges of a cube so that
The sum of the numbers on the edges emanating from each vertex is the same.
The sum of the numbers on the edges lying on each face is the same.
The sum of the numbers on all of the edges is as small as possible.
8
1
10
9
3
12
11
2
5
4
13
6
Solution. We claim that the assignment depicted satisfies the three conditions. We list the
labels:
.
To justify the claim, we make some observations where is the common sum of the edge labels
at each corner, is the common sum of the edges labels of each face, and is the total sum of
the edge labels for the entire cube. Each edge is common to two corners and each edge is
common to two faces. Since there are eight corners, we’d have
, or
. Since there
are six faces, we’d have
, or
. Therefore, is a multiple of both 4 and 3 which
implies is a multiple of 12.
Now, since
, it must be the case that
. From the previous
paragraph, we can conclude that
. The above labeling is an example where
,
, and
. It can be shown using linear algebra that for any assignment satisfying the
first two conditions the sum of the labels on opposite pairs of edges (e.g.
and
) must be
the same. In this case, that sum is 14.