Required Mathematical Intelligence

Chapter 128 Can the Cup Face Down
There are 14 cups on the table, 3 of which are facing up, and now turn 4 cups each time (turn the cup with the mouth up to face down, and turn the cup with the mouth down to face up).Question: Is it possible to put the mouth of the cup downwards after turning it several times?If not, is it only possible to flip 6 each time? 7?

[Answer: Use 1 for the cup with the rim up and -1 for the cup with the rim down.

The initial state is 3 " ", 11 "-", so the product of 14 numbers is -1, and when flipping a cup, it is "turning 1 into -1 or -1 into 1", When flipping a cup, it is equivalent to multiplying the original state by -1.
When flipping the cup n times, it is equivalent to multiplying n "-1"

Therefore, every time an even-numbered cup is flipped, the result that the initial state is "-1" is not changed.

Therefore, flipping 4 cups each time and flipping 6 cups each time cannot change the result of the product being "-1".

So neither can be done.

And every time an odd number of cups are flipped, the result that the initial state is "-1" can be changed.

So flipping 7 cups each time and flipping an odd number of times can do it.

The specific operation is as follows: In the original state, 3 cups face up, and 11 cups face down.

①Turn 2 cups to face up, flip 5 cups to face down,
After flipping, 6 cups face up, and 8 cups face down.

②Turn 3 cups to face up, turn 4 cups to face down,
After flipping, 7 cups face up, and 7 cups face down.

③Turn the 7 cups upwards.

After flipping, at this time, all 14 cups are facing down, and the task is completed. ]
(End of this chapter)