M14-09

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

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

Bunuel · Accepted 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 4. 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

rishi02 · Answer

Note that (from the question), if white wins, white will always have more moves than black.

Win: White. Then, White > Black.

If Black wins, the number of moves of both white and black will be the same.

Win: Black. Then, White (moves) = Black (moves). 
Why? Because, white cannot have the last move and lose the game (from the question). And white always starts so it will be WB WB WB till blacks last move.

If it is a draw, either of these possibilities exists. 
Win: Draw. White > Black or Black = White.

Statement 1: Black lost 5 games. This is false because anyone who plays the last move cannot lose (from the question). IF black lost 5 games, white will have a clear lead of 5 moves. It is only 4 moves from the question.

Statement 2: This is possible. Imagine that they play about 150 games. 146 could be black wins (So black moves will be equal to white). then last 4 can be White wins (White moves > Black moves).

Statement 3: All games are draws with white having the last move in only 4 of them . Possible

Source :BTG

mejia401 · Answer

In order for whites to avoid losing the game, they must have the last move, and therefore, the move must be one more move than the number of black moves. For blacks to avoid losing the game, they only need an equal amount of moves as the number of white moves. However, it's impossible to determine whether blacks won the games or tied them.

Thanks

Thanks

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation. Wow but looks like critical reasoning question in verbal section rather than a quant question!

Heman105 · Answer

In a game of chess, the moves of whites and blacks alternate with  whites having the first move.

Two Possible Scenarios:

i) Starts with White and ends with Black ............... eg: W -> B  or  W -> B -> W -> B 
This means there will be  equal number of black and white moves

ii) Starts with White and ends with White............... eg: W -> B -> W  or  W -> B -> W -> B ->W 
This means there will be  1 more move of White than moves of Black

We are given that  Whites  have made  2319  moves altogether while  Blacks  have made  2315  moves. 
==> The above statement means "There must be some cases where scenario 2 is applicable"

For understanding purpose, let's  assume   There were 5 Games with following scenarios:

Game 1: w->b.......w->b (2309 moves by each)  
(There can be n number of games with multiple scenarios, what we care about is : "Equal number of white and black moves" )]

As we are given that :  "White made additional 4 number of moves" , there can be   4 games  in which White made last move
We can assume following cases:
Game 2: w->b->w
Game 3: w->b->w
Game 4: w->b->w
Game 5: w->b->w->b->w->b->w

We are given that "If in any game the side that made the last move did not lose"
Results of Games :
Game 1: Black Last move = Black did not lose = Black won or Match draw
Game 2: White Last move = White did not lose = White won or Match draw
Game 3: White Last move = White did not lose = White won or Match draw
Game 4: White Last move = White did not lose = White won or Match draw
Game 5: White Last move = White did not lose = White won or Match draw
----------------------------------
Question is asking  "What can be true"   
I. Blacks lost 5 games

--> Black can lose  at most 4  games

II. Blacks won more games than whites

Let's say : Game 1 resulted in win and Game 2,3,4,5 resulted in Draw
We can say that YES blacks won more games than whites ( In our case, Blacks wins =1 and White wins = 0)

III. All games ended in a draw
 --> Game 1,2,3,4,5 can resulted in draw

Hence II and III are possible

shivammahajan · Answer

I agree with explanation.

JIAA · Answer

Hi  Xylan

Can you please assist me on this?

While i do understand from the Q stem that BLACK has lost 4 games since WHITE has 4 points more than BLACK. This information definitely negates the Statement that Blacks lost 5 games.

However, the other 2 Statements are way too confusing or i'm unable to get the logic.

Would really appreciate your help!
Thanks

XyLan · Answer

JIAA  In a game of chess, the moves of whites and blacks  alternate  with whites having the first move.

Two Possible Scenarios:  
 1) Starts with White and ends with Black ............... eg: W -> B or W -> B -> W -> B 
This means there will be an  equal number  of black and white moves  
 2) Starts with White and ends with White............... eg: W -> B -> W or W -> B -> W -> B ->W 
This means there will be  1 more move  of White than moves of Black

We are given that Whites have made 2319 moves altogether while Blacks have made 2315 moves. 
 The above statement means "There must be some cases where scenario 2 is applicable"

We clearly know because of the difference in 4 points:  2319 - 2315 = 4   
  W  could have either  WON  or  DRAW  the 4 matches.
  Inference :  B  could have either  LOST  or  DRAW  the 4 matches. - The maximum number of macthes that can be lost by B:  4    - Read this slowly   
  
For understanding purpose, let's assume: There were 5 Games:  {1, 2, 3, 4, 5}

Question is asking "What  CAN  be true"  MUST " be true] -   Q is looking for a   possibility  , NOT a   certainity   
I. Blacks lost 5 games
 Black can lose AT MOST 4 games

II. Blacks won more games than whites 
Let's say: Game 1 resulted in win and Game 2,3,4,5 resulted in Draw
We  CAN  say that  YES  blacks won more games than whites ( In our case, Blacks wins =1 and White wins = 0) -   A possible scenario

III. All games ended in a draw
 Game 1,2,3,4,5  CAN  result in a draw -   A possible scenario

Hence, ONLY II and III  CAN  be possible.

antartican · Answer

One thing imp in this question is that Black will never exceed White even if it wins.
To Demonstrate:
Black win : W-B-W-B-W-B 
Black Draw : W-B-W-B or W-B-W-B-W
So black is never exceeding white's count in an individual game.
So black can at most lose 4 games. 
Black Can win more than 5 games 
And all games may end in draw.

Please give Kudos if you like my answer

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

darkmaster0099 · Answer

I have a query regarding statement 1.

White has 4 more moves than black.

However, isn't it possible that black could have won 5 matches in the scenario wherein for the rest of the 9 games there were draws with white making the last move?

For instance, say black wins 5 games and is leading by 5 moves. However, for the rest of the 9 games, there were draws with white making the last move. The stem mentions that "the side that made the last move did not lose" - this contains the possibility that one who made the last move could have drawn.

So I don't understand why statement 1 can't be true.

Can anyone please help here?

Thanks!

Bunuel · Answer

The statement 'the side that made the last move did not lose' implies that only the side that made the last move could possibly win. Since whites made 4 more moves than blacks, whites had the last move in 4 games. Thus, whites could have won only those 4 games. In all other games, blacks made the last move and therefore could not have lost in those games.

A_ · Answer

I like the solution - it’s helpful.

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