Math Snake Puzzle

Question

A colleague recently showed me the following puzzle game and I'm interested in how this can be solved. I thought it would be a good talking point for you guys as well :)

A detailed description of the puzzle is here. A sequence of 7 cubes may be rotated about the axis. The puzzle is to rotate them until all 4 equations are correct, such as $2 + 2 / 4 = 1$. Operators are evaluated left to right. The faces of the cube presented in the video (not agreeing with the image) are:

The following pictures show the sides...

Question

Other than trial and error, is there an easy way to solve this?

Answer 1

I just realised that I didn't get it, I just thought all the cubes had to show different numbers/symbols.

A solution that uses each number exactly once, and two different operations $$ \frac{4}{2}+1=3 $$

I found that thinking about what divisions were possible. Obviously we can divide by one, but then we're just left with one operation ($+$, $-$ or $*$) on $\{2, 3, 4\}$, so 9 possibilities, it's not hard to check all of them and conclude that there are no solutions.

If we want the division to give an integer result the only other option is $\frac{4}{2}$, and that gives the solution I've already named.

If we just ignore a remainder from a a division, we get more solutions, like: $$ \frac{4}{3}+1=2 $$

Answer 2

The puzzle in the video only has 1 solution.

$$\begin{array} {|c|c|c|c|c|c|} \hline 4 & - & 3 & \times & 3 & = & 3 \\ 2 & + & 2 & \div & 4 & = & 1 \\ 1 & \times & 4 & - & 2 & = & 2 \\ 3 & \div & 1 & + & 1 & = & 4 \\ \hline \end{array}$$

I doubt there is any significantly faster way to solve it than with a computer. It's only $4^6 \div 4 = 1024$ cases to check. I doubt there is a faster way than with a computer, since in general there can a large number of solutions. This is a combinatorial logic problem, so it is NP, and I don't see any obvious way to put it into P.