marcodonzelli wrote:
In a game of chess the moves of whites and blacks alternate with whites having the first move. During a certain chess tournament whites have made 2319 moves altogether while blacks have made 2315 moves. If in any game the side that made the last move did not lose, which of the following can be true about the tournament?
I. Blacks lost 5 games
II. Blacks won more games than whites
III. All games ended in a draw
A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
M14-09
Official Solution:
In a game of chess, the moves of whites and blacks alternate with whites having the first move. During a certain chess tournament whites have made 2319 moves altogether while blacks have made 2315 moves. If in any game the side that made the last move did not lose, which of the following can be true about the tournament?
I. Blacks lost 5 games
II. Blacks won more games than whites
III. All games ended in a draw
A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
Let's examine each option.
I. Blacks lost 5 games
Since the moves of whites and blacks alternate with whites having the first move, then the fact that the whites made 4 more moves than blacks, means that in only 4 games whites made the last move. Therefore, in all other games, blacks made the last move and since we know that the side that made the last move did not lose, then blacks could not have lost in any of the games but in these 4. So, the maximum number of games black could have lost is 3. Thus, I cannot be true.
II. Blacks won more games than whites
From above, we know that blacks had the last move in all but 4 games. Thus, blacks could not win in only those 4 games but they could have won in all other games. So, could have won more games than whites.
III. All games ended in a draw
The stem does not give any stipulation on draws, so any game, no matter whites had the last move or blacks could have ended in a draw. Hence, all games could have ended in a draw.
Answer: D
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