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16 Sep 2014, 00:52
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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

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16 Sep 2014, 00:52
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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

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.

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25 Feb 2015, 14:12
Bunuel wrote:
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

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.

I dont quite understand how scenario I is impossible. Granted, you are correct when you say that the moves made by the whites should be up by 5, or a difference of 5 (moves white - moves black), but consider a case where the whites won 5 games, but tied an x amount games in a condition where the last move was made by the blacks, the difference of 5 then changes, could be 4 (as in this case) could be less. Is this possible? Does my logic work or am I looking too far into it? Please help!
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14 Jul 2015, 19:00
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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
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16 May 2016, 11:54
I think this is a high-quality question and the explanation isn't clear enough, please elaborate.
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21 May 2016, 12:52
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Hi Experts / Bunuel / chetan2u ,

As mentioned in the question that White made the First move.
Total No of moves made by white- 2319
Total No of moves made by Black- 2315

White made 4 extra moves.

Because White made the first move so, we can take it like this-

WB WB WB WB WB WB WB WB WB WB WB WB WB............

We will get 2315 pairs of WB.
Now, if we talk about those 4 extra moves made by White. 4 games occured in the tournament and the last move was made by WHITE

As per the question "in any game the side that made the last move did not lose".
we can say either all the games were drawn or white won the game.

How could part 2 can be correct (II. Blacks won more games than whites)..??

Thanks and Regards,
Prakhar
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21 May 2016, 13:08
The question asks which of the below statements CAN be true not necessarily true

While we cannot say for sure how many games have been won by Blacks for sure. It is possible that black win more games as they have played 2315 moves. And since white starts game these moves are the last moves for a game. Hence could be possible that Blacks would have won playing last.

SO the answer is D

Hope this helps.
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26 Jun 2016, 05:58
I think this is a high-quality question and the explanation isn't clear enough, please elaborate.
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26 Jun 2016, 06:07
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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
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13 Aug 2016, 05:50
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!
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03 Oct 2016, 01:57
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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) [1 game in which white and black moved equal number of moves = 2309]
(There can be n number of games with multiple scenarios, what we care about is : "Equal number of white and black moves" )]

[ Now, remaining White moves = 10 and Black moves = 6]

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" [ not what "must" 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
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24 Oct 2016, 16:00
hemant89 wrote:
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) [1 game in which white and black moved equal number of moves = 2309]
(There can be n number of games with multiple scenarios, what we care about is : "Equal number of white and black moves" )]

[ Now, remaining White moves = 10 and Black moves = 6]

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" [ not what "must" 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

I think this should be the official explanation. Simply Awesome!
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03 Nov 2016, 02:17
I agree with explanation.
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30 Jul 2017, 02:00
I guess I am missing something here. Why can't there be 28 games with white having last moves in 16 and blacks having last moves in 12? In this case, Black can lose 5 games.
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27 Dec 2017, 06:22
If BLACKS WIN, the number of moves made by blacks and whites will be EQUAL (whites made the first move, blacks made the last)
If WHITES WIN, the number of moves made by whites will ALWAYS BE ONE MORE THAN MOVES BY BLACKS.
If a DRAW happens, two possibilities = either the blacks would draw at their move, thus there will be EQUAL moves OR the whites draw on their move and thus will have ONE MORE MOVE THAN BLACKS.
therefore, the BLACKS CAN NEVER HAVE MORE MOVES THAN THE WHITES (when they win or draw), but CAN HAVE ONE LESS MOVE (when they lose)

I - IMPOSSIBLE. if the blacks lose 5 games, +5 for whites. no matter how many draws happen, the difference can never be less than +5 because when the whites draw, one more move is added to the whites (the difference will be greater than 5) or the blacks draw and the difference stays at 5 (equal moves). but the difference above is 4. therefore the whites have only won 4 games.
II - POSSIBLE, blacks can win many games, say 10, but the moves will stay the same for both. the whites can win only 4 games and give the difference above.
III - POSSIBLE, if whites dragged four games, the difference can be increased by 4. the blacks could have dragged many games but the score stays the same.
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22 May 2018, 13:43
I think this is a high-quality question and the explanation isn't clear enough, please elaborate.
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01 Aug 2018, 13:51
Can there not be more than 5 games in the tournament, example 50 or 100 games. From question stem we can imply that 4 extra matches were won by white, but how do we know that black did not lose 5 games? Sorry if my question sounds silly.

Bunuel
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07 Apr 2019, 04:55
Bunuel wrote:
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

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.

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
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08 Apr 2019, 00:03
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JIAA wrote:
Bunuel wrote:
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

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.

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

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" [ not what "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.
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