In a game of chess the moves of whites and blacks alternate

Question

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

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

Bunuel · Accepted Answer

It talks about the moves players make with black or white figures.

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

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

From the stem it follows that there were only 4 games in which whites had the last move. These 4 games were responsible for the difference in the total number of moves made by whites and blacks during the tournament. We know that these 4 games were not won by blacks (but they could well have ended in a draw). All the other games could have been won by blacks or ended in a draw. Thus, scenarios II and III are possible.

Scenario I is impossible. It means that there were at least 5 games in which whites had the last move. If this were true then the difference between the total number of moves of whites and blacks should be at least 5. In fact, it's only 4.

Answer: D

Bunuel · Answer

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

vscid · Answer

assume that each game is made up of total 2 moves only.
since whites start, we have each game as WB.
so we have 2311 games which have this combination WB.
this means whites have made 2311 moves and blacks also 2311 moves.
now if whites have made 4 more moves than blacks, this means that in 4 games they have made an extra move.since the game starts with whites, this means that the game has also ended with whites. So, we have have 4 games with combination WBW, giving one extra white move per game.
Thus so far we have 2311 games with combination WB and 4 more games with combination WBW. so we have 2315 black moves and 2319 white moves as given.
Thus we could have blacks with more games won, than whites.
Also, its possible that every game ended in a draw as they say that the side that made the last move did not loose, which also means that the side that made the last move, either won or faced a draw.
good one+1

incognito1 · Answer

It's all about counting the number of games that end on either side. 4 games end with white being the last player, and 2315 end with black being the last player.

Let   = number of whites moves,   = number of blacks moves

Consider a single game of chess; since whites start, blacks can either win or draw on their turn. In that case:
  =

Similarly, whites can win or draw on their turn. In that case:
  =   + 1

We're given the sum total of all white and black wins in a tournament. Clearly, whites have only won OR drawn 4 more games (2319 - 2315). Blacks, on the other hand have won OR drawn as many as 2315 games. Using this information, you can now solve the 3 choices.

honchos · Answer

Bunel,

I need your help in understanding the question.

whites: 2319 Moves
Black: 2315 Moves

what dos this means?

Is it talking about chess's black and white square boxes?

dabhishek87 · Answer

I can not be true as if 5 blacks lost then difference between two moves should be alteast 5 or more
II can be true
eg. 1 game won by Black move WB
all other games are tied...as WB or WB...W this can result in the move difference.
III can be true
all games are tied as WB or WB....W or WB....B this can also result in move difference
Answer is D

jkrishh7 · Answer

*Simple Words*
Total White Moves  2319 
Total Black Moves  2314 
As given white made the first move, if we take the difference of  2319-2314=4  means that  upto 2314 moves black made the last move  and since white has 04 more moves, we can safely assume that for 04 games white made the last move and won the game. So remaining games can be anything draw or black won.

Only II & III Possible is D

Kinshook · Answer

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

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

For white to win a game, it must play 1 more move than black since it plays first and last moves.
For black to win a game, it must play equal moves as white since white plays first and black plays last move.
For draw, white plays 1 more move than black or both play equal moves.

Since white made 2319 - 2315 = 4 moves more than black, white wins at most 4 games.

I. Blacks lost 5 games
If black lost 5 games, white won 5 games
CAN NOT BE TRUE

II. Blacks won more games than whites
White wins at most 4 games, rest of the games were draw or black wins.
CAN BE TRUE

III. All games ended in a draw
Since white made 2319 - 2315 = 4 moves more than black, white wins at most 4 games.
It may be possible that all games ended in draw. 
CAN BE TRUE

A. III only 
B. I and II only 
C. I and III only 
D. II and III only 
E. I, II, and III

IMO D

In a game of chess the moves of whites and blacks alternate

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