Spinning Tables

The optimist sees a glass that's half full.
The pessimist sees a glass that's half empty.
The engineer sees a glass that's twice as big as it needs to be.

You're blindfolded, led into a room, and seated in front of a square table. You're told there's a wineglass at each corner, and the wineglasses are randomly up or down.

Your job is to get them all pointing the same direction.

You can tell whether they're up or down by feel, but you can't just feel them all. You have to solve the problem in steps. Once a minute, you can grab any two, and flip the state of either, both, or neither.

You think you've gotten off easy. She's not done.

• Between each step, she'll rotate the table to some random configuration.
• When you get them all in the same state, she'll immediately whisk off your blindfold.

and

• She's naked.

She's doing most of the work and bores quickly. Either find a strategy that guarantees quick, deterministic success or face the prospect of dining alone.

Hat tip to that mountain-climbing bridge player, Rich Laver.