Correct pass required

Question

I was sent to a training centre, for a super secret meeting at 'No-Such-Agency', and arrived in Reception with the orange pass only to be told I must first visit Clearance with the brown pass, and then attend Vetting with the blue pass before returning to Reception with the green pass.

Whoever said that US. Government bureaucracy was easy?

Answer 1

Assuming the following rules:

- The squares on the grid edges are the doors with a pass colour that you need to pass from one room to another. (E.g. You can travel west from Reception with your orange pass, but not south to Vetting.)
- The paired arrows in each cell represent exchanging your pass for a different colour. (So in the room west of Reception you can exchange a green pass for purple or vice-versa.)
- Multiple swaps can be performed in a room. (E.g. In Vetting, you can exchange purple for green and then green for blue.)

First, some observations:

- If a room has only one door of a particular colour, and no exchange-arrows with that colour, then that room represents a dead-end for that pass colour. (E.g. Entering the top-left (north-westernmost) room with a green, brown, purple or blue pass.)

- If a room has only one exchange arrow with a particular colour, and no doors of that colour, then that exchange-arrow can be ignored. (E.g. The top-left room has no orange door, so the yellow-orange exchange is unusable.)

- These rules can be applied iteratively, and a number of paths (and some entire rooms) can be marked as unusable dead-ends. (E.g. In the top-left room, with the yellow-orange exchange marked unusable, the single yellow door is also unusable, and the entire room is a dead-end. Then in the room immediately east, the yellow-green exchange is unusuable, and the remaining doors are all unique colours, so this room is also a dead-end.) In fact, this enables us to ignore over half the rooms when searching for a path through.

With all those options crossed out, we end up with the following map:

Now the unique path that I can find is:

Reception: Orange pass
Go W twice
Swap Orange for Red
Go W thrice
Swap Red for Green
Go S
Go E twice
Swap Green for Blue
Go E
Swap Blue for Brown
Go E
Phew, we made it to Clearance. On to Vetting, which is right next door, but we can't get there with our Brown pass...
So we retrace our steps for a few rooms..
Go W
Swap Brown for Blue
Go W
Swap Blue for Green
Go W twice
Go N (here we diverge)
Swap Green for Purple for Brown
Go W
Go N
Go E
Go N
Swap Brown for Blue for Yellow
Go E
Swap Yellow for Orange
Go S twice
Go E
Go S
Swap Orange for Purple
Go E
Swap Purple for Red for Orange
Go E
Swap Orange for Purple
Go E
Swap Purple for Green for Blue
We made it to Vetting. Not much further now.
Swap Blue for Green
Go N
We are back at Reception with our Green pass.

Map: