Basically, there are 12 items to be ‘added’ to the board one at a time, and what you need to discover is the proper order to add them in.

Most of these items will “grow” in some way when certain of the other pieces are added after they on the board.

The items have varying numbers of ‘growth stages’ they need to go through.

Clearly you need to ‘grow’ each item all the way to its max.

So, for example, if an item must ‘grow’ seven times, it needs to go onto the board before at least seven other items, yes?

But maybe not, because there seems to be other interactions. For example, say you have pieces a, b, c, d, and e on the board. Add f. This causes a, b, c, and e to grow…and then, somehow, this causes a to grow again, and then d. So you get two growth stages in reponse to added f…but maybe only if a and e are already there…
Anyway, I was wondering it there is a logical way of setting out to solve a problem like this?

I’ve been blundering about, and managed once to get 10 out of 12 ‘finished’, but all I’m fairly sure of is which item goes in last.

I think the big question is this: If item b is placed at one point, and placing item c later causes b to grow, does that imply that the ultimate solution requires b to be placed before c? I would expect this is true; otherwise, it seems the puzzle would simply amount to lucking in to the solution (unless instead the solution somehow depends on the graphical representations of the objects).

In that case, the best approach would be to make a random attempt at first, recording the growth dependency.

For example, just now I tried putting in the items as they are ordered down the left column. When I put the third item (the pinwheel) on, the second (gear) grew. If my expectation is correct, this means gear < pinwheel (any correct order must place the pipe before the pinwheel).

Keep track of this, and use transitivity whenever available. In other words, if you later determine pinwheel < egg, then from above we know gear < egg.

Eventually this will refine to a linear order (all twelve will be lined up in order) and you got your solution.

Of course, that should read: For example, just now I tried putting in the items as they are ordered down the left column. When I put the third item (the pinwheel) on, the second (gear) grew. If my expectation is correct, this means gear < pinwheel (any correct order must place the gear before the pinwheel).