A new six-sided die

Question

Enrico wants to construct a new six-sided die with the following properties:

• Each side should show a positive integer
• Different sides should show different integers
• If two sides share an edge, then their integers should differ by at least 2.

What is the smallest possible sum of the six integers on Enrico's die?

Answer 1

As already mentioned, the smallest possible sum is 27, resulting from the numbers 1, 2, 4, 5, 7, 8 arranged like this:

 1
4758
 2

Optimality is a consequence of the observation that no three sides of the die may show three consecutive numbers (e.g. 1, 2, 3 or 3, 4, 5 are not possible).

To see this, suppose that $n$, $n+1$ and $n+2$ were all on different sides of Enrico's die. Then because of rule #3, $n$ and $n+1$ must be located on opposite sides, and so must $n+1$ and $n+2$: a contradiction.

The fact that no three sides may show three consecutive numbers implies that there must be at least two gaps in the number sequence between three pairs of consecutive numbers, and we see that the sum of the sequence is minimized for gaps at 3 and 6.

Answer 2

Can I use a barrel die? Then we could do an arrangement of

1 3 5 2 4 6

for a sum of

21.

We can also do

1 5 3 6 2 4

which works even better, since opposing faces have the same sum.

Answer 3

I was able to get a sum of 27. The three pairs of opposite sides are 1 and 2, 4 and 5, and 7 and 8.