martin gardner four glasses puzzle

You’re blindfolded and sitting before a lazy susan. On each corner is a glass. Some are right side up and some upside down. On each turn you can inspect any two glasses and, if you choose, reverse the orientation of either or both of them. After each turn the lazy susan will be rotated through a random angle. When all four glasses have the same orientation, a bell will sound. How can you reach this goal in a finite number of turns?

• Grab two diagonal glasses. Turn them both down.
• Grab two adjacent glasses. You are guaranteed to have grabbed one glass from step 1 and one glass you've never grabbed. If the new glass is up, turn it down.
• You are now guaranteed that at least three glasses are down. If they are all down you win. So let's assume that one is up: DDDU
• Grab two diagonal glasses. If you happen to grab the one up glass, you can flip it down and you win. So let's assume you grab the two down glasses that are adjacent to the up glass. Flip one of them up. Now you have DDUU (or UDDU, which is cyclically equivalent).
• Grab two adjacent glasses. If they are the same orientation, flip them, and you win. So let's assume they are different orientations. Flip both of them and you have (DUDU).
• Grab two diagonal glasses. They are guaranteed to be the same orientation. Flip them so they match the other two.

• I first realized you have to alternate between diagonal and adjacent grabs. If you stay with one type, you're not guaranteed to make progress.
• I next realized that after two rounds you are guaranteed to have at least 3 cups with the same orientation.
• I next realized that you don't actually need to know that you've won. You just need there to be a winning state at some point during all your flips.
• I next realized if you have DUDU, you are one step away from winning.
• I next realized you can get to DDUU from DDDU. And then you can get to DUDU from DDUU.
• Question: Is there a procedure to verify that you've won? What procedure will guarantee that you win and that you know that you win? For example, this procedure will not let you know that you've won if you win in step 3.