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# Rita and Sam play the following game with n sticks on a

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

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03 Apr 2012, 12:02
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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. 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
[Reveal] Spoiler: OA
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Re: Rita and Sam play the following game with n sticks on a [#permalink]

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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.

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

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31 May 2013, 05:22
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE
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Re: Rita and Sam play the following game with n sticks on a [#permalink]

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03 Jun 2013, 20:57
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.

Hi Bunnel,

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 [#permalink]

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04 Jun 2013, 02:46
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Expert's post
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.

Hi Bunnel,

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 [#permalink]

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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.

Hi Bunnel,

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 [#permalink]

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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.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 07 Apr 2012 Posts: 126 Location: United States Concentration: Entrepreneurship, Operations Schools: ISB '15 GMAT 1: 590 Q48 V23 GPA: 3.9 WE: Operations (Manufacturing) Followers: 0 Kudos [?]: 10 [0], given: 45 Re: Rita and Sam play the following game with n sticks on a [#permalink] ### Show Tags 02 Sep 2013, 20:05 so what is the generalisation in such questions or we just have to analyze everytime? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7118 Location: Pune, India Followers: 2128 Kudos [?]: 13621 [0], given: 222 Re: Rita and Sam play the following game with n sticks on a [#permalink] ### Show Tags 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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02 Sep 2013, 20:39
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 [#permalink]

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03 Sep 2014, 14:09
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20 Oct 2015, 15:49
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Re: Rita and Sam play the following game with n sticks on a [#permalink]

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25 Nov 2016, 06:46
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Re: Rita and Sam play the following game with n sticks on a   [#permalink] 25 Nov 2016, 06:46
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