8.24 On a circle, there are several black and white chess pieces. Two people, A and B, play a game: A takes away all black pieces that have adjacent white pieces, then B takes away all white pieces that have adjacent black pieces. They take turns doing this until only one color of pieces remains.
(1) Suppose there are 40 pieces on the circle at the beginning. Can it be possible that after each person takes twice, only one piece remains on the circle?
- (2) Suppose there are 1000 pieces on the circle at the beginning. What is the minimum number of steps required to leave only one piece on the circle?
[Solution] The right diagram shows 41 pieces, of which 29 are black and 12 are white. The numbers marked beside the pieces indicate how many steps before the end the piece was taken. Pieces without numbers should all be marked with 4, which is omitted in the diagram. It is easy to verify that after each of the two players takes two pieces, only one black piece marked with 0 remains. This indicates that the answer to question (1) is affirmative, because we just need to take the black piece marked with an arrow.

Now let's analyze the process of taking pieces in reverse. Since only one piece remains on the circle at the end, let's assume it is a black piece. Therefore, in the previous step, at most two white pieces were taken, meaning there were at most 3 pieces on the board. Going back one more step, at most 4 black pieces were taken, meaning there were at most 5 black pieces and 2 white pieces, totaling 7 pieces on the circle. We use $b_{t}$ and $w_{t}$ to represent the maximum number of black and white pieces on the circle when we backtrack $t$ steps. It is easy to see that when $b_{t} > w_{t}$, $b_{t+1} = b_{t}$ and $w_{t+1} = w_{t} + 2b_{t}$; when $b_{t} < w_{t}$, $b_{t+1} = b_{t} + 2w_{t}$ and $w_{t+1} = w_{t}$. Using this recursive formula, we can sequentially derive the maximum number of black and white pieces and the total number of pieces on the circle, as listed below:

From the table, we can see that to leave only one piece on the circle after 7 steps, there can be at most 577 pieces on the circle at the beginning. Therefore, when there are 1000 pieces on the circle, it will take at least 8 steps to leave one piece.

Such an example can be constructed. Let's go back to the initial example, which is already backtracked 4 steps. Then, place one white piece on each side of every black piece, and then place one black piece on each side of every white piece, and repeat this process twice, to get the scenario of backtracking 8 steps with the maximum number of pieces, i.e., 408 white pieces and 985 black pieces, totaling 1393 pieces on the circle. Note that in the 8th step, 816 black pieces were added, and by removing 393 of these black pieces, we get an example with 1000 pieces that meets the requirements. In summary, when there are 1000 pieces on the circle, it will take at least 8 steps to leave one piece.