The Cabbage, Goat, and Wolf problem intrigued me the other day too... so I did a fantasy world. There aren't incompatible pairs but there are limitations on what can be done.

Imagine a universe where there are three types of people: wizards, warriors, and priests. Wizards can open a portal that allows two people to go through at a time, but they cannot go through the portal themselves. Priests can summon people from other locations to their location or teleport to the location of another person. Warriors cannot teleport or summon, but may be teleported or summoned by others.
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Given four wizards, a priest, and a warrior - what are the necessary steps to move them all to a new location?

To which GPT-4 replied:

In order to move everyone to a new location, we'll need to use both the powers of wizards and the priest. Let's denote the wizards as W1, W2, W3, W4, the priest as P, and the warrior as Wr. We also need to define two locations: the starting location (Location A) and the destination (Location B). Here is a possible sequence of steps:

1. W1 opens a portal from Location A to Location B. W2 and W3 go through this portal.

(Location A: W1, W4, P, Wr; Location B: W2, W3)

2. W1 opens another portal from Location A to Location B. This time W4 and Wr go through.

(Location A: W1, P; Location B: W2, W3, W4, Wr)

3. Now the priest (P) uses his summoning ability to bring W1 to Location B.

(Location A: ; Location B: W1, W2, W3, W4, Wr)

4. Finally, the priest (P) uses his teleportation power to move himself to Location B.

(Location A: ; Location B: W1, W2, W3, W4, P, Wr)

Everyone is now at the new location, Location B.

I found that rather interesting - especially the short hand for being able to write about it.

I then revised it (because I didn't like the second step of the solution):

Redo this exercise with the limitation that a wizard may only open one portal.

And then it went on with:

The limitation that a wizard can only open one portal makes this a bit more tricky. But, it's still possible to move everyone using the powers of both wizards and the priest. Let's again denote the wizards as W1, W2, W3, W4, the priest as P, and the warrior as Wr. Here is a possible sequence of steps:

...

The thing I like about this is that the pattern of this problem doesn't even exist anywhere else (I also did another one with Wizards, Warriors, and Witches to see how it would tweak things if all the roles started with a 'W') and then changed the witches power to be a swap instead of a summon.

I personally found this more convincing than the goat, cabbage, wolf.

This was partly inspired by the old school SHRDLU world where shapes and colors and things stacked on top of each other need to be manipulated. And in doing that example the world of SHRDLU is described in natural language. And while SHRDLU was impressive in itself (and over 50 years old) - it was programmed to manipulate that world and the program wasn't useful for abstracting to other worlds. GPT-4 was impressive at being able to manipulate the world without needing to be programmed for that world (and you could then expand it to other things that it is less familiar with).

Imagine a world that is made up of spheres, cones, bowls. An object has a size and a color.
There is a small red sphere in a medium blue bowl.
There is a large red cone.
There is a small green bowl.
There is a medium blue sphere in a large red bowl.
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What are the steps necessary to move the red sphere into the red bowl?

To which it replied:

...

It's important to note that these steps assume you're allowed to place the red sphere in the red bowl even though there's already a medium blue sphere there. If that's not the case and you're required to empty the bowl first, then the steps would be a little different.

And I then asked it:

What steps are necessary if the bowl must be empty first?

And got back a response that included:

Temporarily place the small red sphere in a safe location, for instance, inside the small green bowl, making sure it won't roll away or get damaged.

Again, I find this more impressive than a reformulation of a well known problem as there's also some implied understanding of the world in it (spheres can roll away unless put in a bowl).