Equally likely outcomes, probability question

If the team is equally likely to win as to lose each game,
what is the probability that they win a string of at least
6 games in a row?

My thinking in this was just counting up the occurrences of the team winning 8 games in a row, which I figured could only occur 6 times out of the possible 256 outcomes. 256 because 2 to the 8th. That guess was wrong. Is there something I am doing wrong. Please be specific in your response.

8 games: 1/2^8 = 1/256

7 games (exactly) 1/2^8 times 2 = 1/128
The 8 is because you need to ensure the 8th game is a loss.
The 2 is because the loss can come first or last

6 games (exactly):

Case 1: The first 6 games (OR the last 6 games) are won
So the 7th game (or the 2nd game) must be lost, no-one cares about the result of the 8th game (or the 1st game)
So 1/2^7 times 2 = 1/64

Case 2: Loss, 6 wins, loss
1/2^8 = 1/256

Answer:
1/256 + 1/128 + 1/64 + 1/256
= 1/32

Where did you get the 8 from? Nothing mentioned in the question about 8 games to play.

There can be 2^6 = 64 outcomes in 6 games and among all of those only 1 outcome where they win all the games.

So, the probability is 1/64.