[namjh]

In reality, the entire map should be sent first to the verifier (with colors hid behind the post-it) so if it was a bogus randomly colored map, you may find two adjacent points having same color if you try extremely many(think hundreds or thousands in the website's case) times. If you sufficiently try many times and fail to find the adjacent points, you will convince that the prover have the map which is correctly 3-colored.

Note that the entire map is sent again, shuffled its color each time after you choose the two points.