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