The Gear Octahedron puzzle actually has $1,327,104$ states. On my website I have a page about the Gear Mastermorphix, which is an equivalent puzzle (same mechanism, but with a tetrahedral outer shape). On that page there is an explanation as to how that number comes about:
$4!$ permutations of the triangular face centres (one of them is held steady and ...
The (twice) given geometrical answer is very clever.
Here's a much less clever solution, using very basic engineering only. It utilises the fact that the diagonal happens to be the longest dimension of the brick.
Stand two of the bricks up on the ground with some room in between. Keeping one corner always touching one brick,
Now you can easily measure ...
Here is my alternative answer that uses wedges:
This was the answer I had in mind when I wrote the puzzle.
Congratulations to @FireCase for solving the puzzle and showing there are at least two solutions.
I'll start with a couple of assumptions:
we cannot rearrange the pieces between cuts (confirmed by OP in the comments)
each guest needs to get the same amount of cake and frosting, else the split isn't equal.
With these assumptions, we are best off by keeping each cut vertical, and running through the center of the cake.
So let's see what we can get. ...
Why is this enough? Consider the fractions I mentioned. Because of the halving and thirding cuts made in two different perpendicular planes, it will be possible to either divide numerators or multiply denominators by 2, 3 or both (if needed). That enables us to make 7*1/7, 8*1/8 and 9*1/9 according to the needs.
Obviously, the planes can be ...