Riddles, brain puzzles and mathematical problems

[MyRedNeptune]

You are suddenly abducted by aliens while sleeping. You wake up in a room and you see an indescribable creature which you soon realise, that somehow, it knows and can speak English. Before you ever have the chance to react the alien makes things clear:
" We were looking for other intelligent life in the universe and we finally found Earth. Humans have confused us in a great extent. We want to measure human intelligent and we abducted a human being randomly (that happened to be you). You see if a civilisation is not smart enough then it must be destroyed. If you manage to solve the intelligence test we have prepared for you then we will let you free and leave your planet in peace. However if you fail, your civilization probably isn't smart enough."

"This has to be a nightmare" you think, but then the alien continues:

"Next to you there is another room which you can't see or interact in any way. In this room there is a square table with 1 card on it in each corner (total of 4 cards). You don't know which cards are face up and which are face down. Your goal is to turn all the cards face up by following the following rules:
At each phase you tell me which card or cards you want me to turn. Then I go to the room and turn the cards you told me. However in the end of the phase I am going to rotate the table as many times as I want to make it more difficult. If all the cards after turning them as you told me are face up then you win and you will be set free. If not, then you try again with the current position of the cards. You have 3 hours to solve the test."

How are you going to solve the problem and save the Earth?

Spoiler below!

Because the table is rotated an unknown number of times after each turn, the absolute position of each card is impossible to predict. Therefore, for the sake of simplicity, we can view our table as stationary and our choices of flipping cards as falling randomly on the table. It is the position of the cards relative to each other that is important. There are 5 such patterns possible, and they are each assigned a number in the chart below (the dark circles represent cards that are face up and the win state is unlisted):



There are also 5 possible moves we can make, and they coincide with the states above. For example, 1 is an inversion - all cards are flipped, 2 is flipping two cards on opposite corners, etc. Now I will make a list of ways in which each move can affect each state.


state/move + move/state = possible state(s)

1 + 1 = win
1 + 2 = 2
1 + 3 = 3
1 + 4 = 4
1 + 5 = 5
2 + 2 = win, 1
2 + 3 = 3
2 + 4 = 4, 5
2 + 5 = 4, 5
3 + 3 = win, 2, 1
3 + 4 = 4, 5
3 + 5 = 4, 5
4 + 4 = 1, 3, 2
4 + 5 = win, 3, 2
5 + 5 = 1, 3, 2


Using this list, we can now create a strategy for solving the puzzle. It will be written in the following format:

(possible states);
move (possible states);
move (possible states);
...etc

So, the semicolon marks the beginning of a new turn, the first number of each line is the move we make in the current turn, and the numbers in the parentheses indicate the possible states after the move. Each new turn is given with the assumption that the previous turn did not result in a win state.

(1, 2, 3, 4, 5);
1 (win, 2, 3, 4, 5);
2 (win, 1, 3, 4, 5);
1 (win, 3, 4, 5);
3 (win, 1, 2, 4, 5);
1 (win, 2, 4, 5);
2 (win, 1, 4, 5);
1 (win, 4, 5);
5 (win, 1, 2, 3);
1 (win, 2, 3);
2 (win, 1, 3);
1 (win, 3);
3 (win, 1, 2);
1 (win, 2);
2 (win, 1);
1 (win)

I should be given the title of Earth's Champion because I solved it in 30 minutes.