There is an easy proof that you can't do better than 45. There are 54
surface squares. There are six 0's on the domines, which must
mismatch. You also get a mismatch from each of the 1-6, 2-5, and 3-4
dominoes. That is 9 mismatches, so you cannot beat 45.
With a little work, you can improve that to 44:
Consider the dominoes that fold over an edge of the die. How many are
there for each face of the die?
Each face has 4 dominoes that fold over its edges so that the domino can
score both squares. It also needs another one so that
it has an even number of squares covered by its own private dominoes.
Look at the two dominoes on a particular face. One covers the center
square and an edge square; the other covers a corner square and an edge
square. So the five dominoes that are shared with other faces cover
three corner squares and two edge squares. That means that on the
entire figure, there are eighteen corner squares covered by dominoes
that fold around edges. But there are only eight corners on the die, so
there can only be sixteen corners squares covered that way.
This means that 45 is not an attainable score.
I unfold the die to get a picture like this:
(view with a fixed-width font)
3
1 5 6 2
4
That will be used to assign numbers to the dominoes later.
I will look for a solution that has no domino spanning the 1-2 edge, and
has two dominoes spanning the 3-5 and 4-6 edges. The others edges each
hae one domino.
After a litte trial and error, I found this arrangement:
(each letter identifies the 2 squares of a domino that folds over and
edge)
amk
---
-de
a-hhde--k-m-
-----cc-----
-l-i--fgjj-b
fg-
---
ilb
The other spaces on faces can be divided into two or three dominoes as
appropriate.
The x-x and 0-x dominoes go on face x. The x-y dominoes are folded
across the x-y edge when possible.
The leaves the 1-2, 1-6, 2-5, and 3-4 dominoes to assign to the 1 face,
the 2 face, the 3-5 edge, and the 4-6 edge
in a way that manages to match one of the numbers. One such way is
this:
1-2 on the 1 face
1-6 on the 4-6 edge
2-5 on the 2 face
3-4 on the 3-5 edge
This scores
12 for the x-x dominoes
6 for the x-0 dominoes
22 for the 11 edge-matching dominoes
4 for the last 4 dominoes.
for as total of 44.
-jim