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Rita and Sam play the following game with n sticks on a table. Each mu

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Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16

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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 04 Jun 2013, 20:26
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eybrj2 wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1,2,3,4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. Tha one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16


I would like to point out one thing about these questions based on games. These games are made to have a sure shot winner (if both players play intelligently and to win) under certain conditions. If A and B are playing, B's move will be decided by A's move if B has to win i.e. there are complementary moves. For example, in this question, if A picks 2 sticks, B must pick 4 sticks. If A picks 3 sticks, B must pick 3 too. So to solve these questions you need to find this particular complementary relation.

This question tell us that one can pick 1/2/3/4/5 sticks. This means n must be greater than 5 to have a game else the one who picks first will pick all and win. If n = 6, the first one to pick must pick at least 1 and at most 5 sticks leaving anywhere between 5 to 1 sticks for the other player. The other player will definitely win. If n= 7, the first player will pick 1 and leave the other player with 6 sticks. The first player will win. So the object of the game is to leave 6 sticks for your opponent. If the number of sticks is a multiple of 6, you can always make a complementary move to your opponent's move and ensure that you leave your opponent with 6 sticks. For example, if your opponent picks 1 stick, you pick 5, if he picks 2 sticks, you pick 4 and so on.

So when Rita starts, Sam can complement her move each time and leave her with 6 sticks at the end if the total number of sticks is a multiple of 6. There is only one multiple of 6 in the options.
Hence, answer must be (D)
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 03 Apr 2012, 12:21
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eybrj2 wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1,2,3,4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. Tha one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16


If the number of sticks on a table is a multiple of 6, then the second player will win in any case (well if the player is smart enough).

Consider n=6, no matter how many sticks will be removed by the first player (1, 2, 3 ,4 or 5), the rest (5, 4, 3, 2, or 1) can be removed by the second one.

The same for n=12: no matter how many sticks will be removed by the first player 1, 2, 3 ,4 or 5, the second one can remove 5, 4, 3, 2, or 1 so that to leave 6 sticks on the table and we are back to the case we discussed above.

Answer: D.
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 03 Jun 2013, 20:57
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Bunuel wrote:
eybrj2 wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1,2,3,4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. Tha one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16


If the number of sticks on a table is a multiple of 6, then the second player will win in any case (well if the player is smart enough).

Consider n=6, no matter how many sticks will be removed by the first player (1, 2, 3 ,4 or 5), the rest (5, 4, 3, 2, or 1) can be removed by the second one.

The same for n=12: no matter how many sticks will be removed by the first player 1, 2, 3 ,4 or 5, the second one can remove 5, 4, 3, 2, or 1 so that to leave 6 sticks on the table and we are back to the case we discussed above.

Answer: D.



Hi Bunnel,
Please explain this:

N = 12, here 1 and 2 shows steps in a game: rita picks 5 first, out of remaining 7 sam can pick a maximum of 5, which leaves 2 sticks after round one. On her next chance rita can pick 2 and win.

R S
1 5 5
2 2 > Rita wins

similarly:
R S
1 4 5
2 3 > Rita wins
R S
1 2 5
2 5 > Rita wins
R S
1 2 2
2 5 3 > Sam wins
R S
1 2 3
2 5 2 > Sam wins

So both can win when n=12.
I agree for n=6, but not for n=12.
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 04 Jun 2013, 02:46
1
cumulonimbus wrote:
Bunuel wrote:
eybrj2 wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1,2,3,4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. Tha one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16


If the number of sticks on a table is a multiple of 6, then the second player will win in any case (well if the player is smart enough).

Consider n=6, no matter how many sticks will be removed by the first player (1, 2, 3 ,4 or 5), the rest (5, 4, 3, 2, or 1) can be removed by the second one.

The same for n=12: no matter how many sticks will be removed by the first player 1, 2, 3 ,4 or 5, the second one can remove 5, 4, 3, 2, or 1 so that to leave 6 sticks on the table and we are back to the case we discussed above.

Answer: D.



Hi Bunnel,
Please explain this:

N = 12, here 1 and 2 shows steps in a game: rita picks 5 first, out of remaining 7 sam can pick a maximum of 5, which leaves 2 sticks after round one. On her next chance rita can pick 2 and win.

R S
1 5 5
2 2 > Rita wins

similarly:
R S
1 4 5
2 3 > Rita wins
R S
1 2 5
2 5 > Rita wins
R S
1 2 2
2 5 3 > Sam wins
R S
1 2 3
2 5 2 > Sam wins

So both can win when n=12.
I agree for n=6, but not for n=12.


That;s not correct.

Both players can win BUT if the number of sticks on a table is a multiple of 6, then the second player will win in any case IF the player is smart enough.

n=12: no matter how many sticks will be removed by the first player 1, 2, 3 , 4 or 5, the second one can remove 5, 4, 3, 2, or 1, RESPECTIVELY so that to leave 6 sticks on the table.
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 04 Jun 2013, 18:12
If the number of sticks on a table is a multiple of 6, then the second player will win in any case (well if the player is smart enough).

Consider n=6, no matter how many sticks will be removed by the first player (1, 2, 3 ,4 or 5), the rest (5, 4, 3, 2, or 1) can be removed by the second one.

The same for n=12: no matter how many sticks will be removed by the first player 1, 2, 3 ,4 or 5, the second one can remove 5, 4, 3, 2, or 1 so that to leave 6 sticks on the table and we are back to the case we discussed above.

Answer: D.[/quote]


Hi Bunnel,
Please explain this:

N = 12, here 1 and 2 shows steps in a game: rita picks 5 first, out of remaining 7 sam can pick a maximum of 5, which leaves 2 sticks after round one. On her next chance rita can pick 2 and win.

R S
1 5 5
2 2 > Rita wins

similarly:
R S
1 4 5
2 3 > Rita wins
R S
1 2 5
2 5 > Rita wins
R S
1 2 2
2 5 3 > Sam wins
R S
1 2 3
2 5 2 > Sam wins

So both can win when n=12.
I agree for n=6, but not for n=12.[/quote]

That;s not correct.

Both players can win BUT if the number of sticks on a table is a multiple of 6, then the second player will win in any case IF the player is smart enough.

n=12: no matter how many sticks will be removed by the first player 1, 2, 3 , 4 or 5, the second one can remove 5, 4, 3, 2, or 1, RESPECTIVELY so that to leave 6 sticks on the table.[/quote]

got it. thanks.
is this gmat question ?
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 02 Sep 2013, 20:05
so what is the generalisation in such questions or we just have to analyze everytime?
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 02 Sep 2013, 20:28
ygdrasil24 wrote:
so what is the generalisation in such questions or we just have to analyze everytime?


To have a sure shot winner, you need complimentary moves. You have to analyze to figure out the complimentary move every time, of course.
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 02 Sep 2013, 20:39
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ygdrasil24 wrote:
so what is the generalisation in such questions or we just have to analyze everytime?


The trick is to rephrase the question in more general terms. In this case it would be: What is the number that can always be divided into even number of times when each division can be up to 5. The answer is one greater than 5 which is 6 because whatever be the first value chosen, the second value can be chosen such that 6 can always be divided into two. The same idea can be extended to the multiples of 6 such that they can always be divided even number of times given that each division can be from 1 to 5.
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 22 Sep 2019, 23:07
Are 1,2,3,4,5 number on sticks or number of sticks?
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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 16 Oct 2019, 13:30
Total number of the sticks must be a multiple of 6. If Rita removes n(<6) sticks in first turn, Sam will remove 6-n sticks in his first turn. This process can goes on until no stic remains on the table. Number of sticks remains after each turn of the Sam is a multiple of 6. Hence it's the Sam's turn after which there is no stick left on the table (as 0 is multiple of 6).




eybrj2 wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16

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Re: Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 17 Dec 2019, 19:49
1
1
eybrj2 wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7
B. 10
C. 11
D. 12
E. 16

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Let’s go through the choices.

A. 7

Since Rita goes first, she can remove only 1 stick. (If she removes any other number of sticks, then Sam will be able to win on his next turn.) Then there are 6 sticks left, and no matter how many sticks Sam removes, Rita will always win (for example, if Sam removes 2 sticks, Rita can then remove 4 sticks for the win). Since we want Sam to win, choice A is not the correct answer.

B. 10

Since Rita goes first, she can remove 4 sticks. Then there are 6 sticks left, and no matter how many sticks (up to the 5-stick limit) Sam removes on his turn, Rita will win on her next turn. Thus, there is no way Sam can win.

C. 11

Since Rita goes first, she can remove 5 sticks. Then there are 6 sticks left, and just as in the explanation for answer choice B, there is no way Sam can win.

D. 12

Since Rita goes first, she can remove any number of sticks, from 1 to 5 (inclusive). Then Sam can remove a number of sticks so that there are 6 sticks left. For example, if Rita removes 2 sticks, Sam should remove 4 sticks so that there are 6 sticks lefts. Once there are 6 sticks left and since it’s now Rita’s turn, no matter how many sticks she removes, Sam will always win (for example, if Rita removes 5 sticks, Sam can then remove 1 stick for the win).

Answer: D
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Please help me to get an answer  [#permalink]

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New post 12 Feb 2020, 05:40
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?
7
10
11
12
16

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New post 12 Feb 2020, 06:10
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If there are 6 sticks on the table before a turn, then after that turn, there will be between 1 and 5 sticks on the table, and the next player can win. So if Sam can, no matter how Rita plays, force Rita to chose from 6 sticks at some point, Sam can always win (assuming he plays perfectly). No matter how many sticks Rita chooses on her first turn, Sam can always choose a specific number of sticks so that, combined, they remove exactly six sticks from the table (if Rita removes one, Sam can remove five, if Rita removes two, Sam can remove four, and so on). So if there are 12 sticks to begin with, no matter how many Rita removes on her first turn, Sam can remove enough sticks on his turn so there are 6 sticks left on the table. Then Sam can always win. So the answer is 12.

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New post 12 Feb 2020, 06:11
Poornimayashas wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?
7
10
11
12
16

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Poornimayashas

I believe D should be the answer.

My reasoning :-
Anyone who can manage to put 6 sticks on the table after picking with his/her chance will win since the other has to pick one on his turn and the rest can be taken by the other person and than that will win.

For 7 - Rita can pick 1 - 6 will remain and Sam will have to pick atleast 1 and Rita can pick the rest to win
Similarly for 10 - Rita can pick 4
For 11 Rita can pick 6 in her first chance.
But for 12 - Rita has to pick atleast 1 in which case Sam can pick 1-5 sticks for the number of remaining sticks on the table to be 6 and he will win with the next chance.

Answer - D
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New post 12 Feb 2020, 06:32
I still didn't understand. This is not a probability question to guess how many they will pick at a time. It's 1,2,3, 4 or 5. Confusing!

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New post 12 Feb 2020, 07:35
Poornimayashas wrote:
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?
7
10
11
12
16

Posted from my mobile device


Merging topics. Please check the solutions above. Hope it helps.
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Rita and Sam play the following game with n sticks on a table. Each mu  [#permalink]

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New post 12 Feb 2020, 13:14
Poornimayashas wrote:
I still didn't understand. This is not a probability question to guess how many they will pick at a time. It's 1,2,3, 4 or 5. Confusing!

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Hi - This is a reasoning question.

Question:
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?


Explanation:

Let us first decide the winning strategy for Rita:
In order to win, Rita (R) must be able to pick the last stick.

In order to decide the winning strategy, Rita must be able to make something constant in the game based on which he would do her calculations.

For any number of sticks that Sam picks, say k, Rita can always pick (6 – k) sticks in her next pick; thus maintaining the sum of the number of sticks as {k + (6 – k)} = 6, i.e. a constant.
For example: If Sam picks 1, Rita can pick 6 – 1 = 5; if Sam picks 2, Rita can pick 6 – 2 = 4; and so on ... if Sam picks 5, Rita can pick 6 – 5 = 1.

Thus, if the number of sticks is 52 (let us assume any random number to see what happens), we would have: We remove as many multiples of 6 as possible from 52, we would be left with 4 sticks. Rita needs to pick all 4 of these in the beginning so that she can finish the game. The sequence in which the sticks are picked would have been:

Attachment:
11.JPG
11.JPG [ 17.5 KiB | Viewed 3266 times ]



Now, let us decide: how can Rita lose? This can only happen if the total number of sticks is a multiple of 6. In that case, Rita HAS to pick some number in the first draw; and then, Sam will use the above strategy and ensure Rita loses. The diagram below explains this situation:

Let the number of sticks be 24 (a multiple of 6) and Rita picks k in the first attempt:

Attachment:
111.JPG
111.JPG [ 13.88 KiB | Viewed 3262 times ]



The only multiple of 6 in the options is 12

Answer D
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