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Math Expert V
Joined: 02 Sep 2009
Posts: 58340
A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 28% (02:33) correct 72% (03:03) wrong based on 50 sessions

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A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

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Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Bunuel wrote:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

The bug is a one vertex of the cube. So remaining vertex to be visited 7. So the bug has to visit a new vertex in each of the the subsequent moves. When he is at a vertex he can select one of the edges in 3 ways.

So probability of selecting an edge = 1/3

So the probability that the bug will visit the every vertex once in the next seven moves = $$(1/3)^7$$ =1/2187
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Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Bunuel wrote:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

total paths for the bug to move 3^7 = 2187
no of ways bug can move = 3*3*2 = 18
so P ; 18/2187 ; 2/243
IMO C
Intern  B
Joined: 11 May 2018
Posts: 24
Location: India
Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Archit3110 wrote:
Bunuel wrote:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

total paths for the bug to move 3^7 = 2187
no of ways bug can move = 3*3*2 = 18
so P ; 18/2187 ; 2/243
IMO C

Pls explain highlighted
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Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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1
Archit3110 wrote:
Bunuel wrote:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

total paths for the bug to move 3^7 = 2187
no of ways bug can move = 3*3*2 = 18
so P ; 18/2187 ; 2/243
IMO C

Pls explain highlighted

draw a cube ; so as to cover max distance in a cube i.e opposite sides diagonal side the bug will start from one of the vertices it can move in 3 directions ; then bug can move in 2 directions ; after which again bug will have 3 options to move
so total max ways a bug can move w/o coming back to vertice already crossed is 3*2*3 ; 18
Intern  B
Joined: 10 Dec 2016
Posts: 21
Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Hi Bunuel,

Please provide the official answer and the explanation.

Thanks
Sekhar M
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Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Total paths available for the bug= 3 per vertex^7moves = 3^7
Favorable paths in order to visit each vertex in these seven moves= 3*2*2*1*1*1*1 = 12
First vertex= 3 options
Second vertex= 2 options
Third vertex= 2 options
Fourth to seventh vertex= 1 option each

Probability = 12/3^7 = 4/729 is the answer IMO, but none of the options have this ans choice.
Manager  B
Joined: 11 Aug 2017
Posts: 60
Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Archit3110 wrote:
Bunuel wrote:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

total paths for the bug to move 3^7 = 2187
no of ways bug can move = 3*3*2 = 18
so P ; 18/2187 ; 2/243
IMO C

the number of ways bug can move ..can you explain the movement keeping in mind the condition stated?? the way you got 18 is not clear to me.
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Joined: 18 Aug 2017
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Concentration: Sustainability, Marketing
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A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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1
draw a cube ; so as to cover max distance in a cube i.e opposite sides diagonal side the bug will start from one of the vertices it can move in 3 directions ; then bug can move in 2 directions ; after which again bug will have 3 options to move
so total max ways a bug can move w/o coming back to vertice already crossed is 3*2*3 ; 18

Mohammad Ali Khan wrote:
Archit3110 wrote:
Bunuel wrote:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

(A) 1/2187
(B) 1/729
(C) 2/243
(D) 1/81
(E) 5/243

total paths for the bug to move 3^7 = 2187
no of ways bug can move = 3*3*2 = 18
so P ; 18/2187 ; 2/243
IMO C

the number of ways bug can move ..can you explain the movement keeping in mind the condition stated?? the way you got 18 is not clear to me.
Manager  B
Joined: 05 Oct 2017
Posts: 79
A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Archit3110 wrote:
draw a cube ; so as to cover max distance in a cube i.e opposite sides diagonal side the bug will start from one of the vertices it can move in 3 directions ; then bug can move in 2 directions ; after which again bug will have 3 options to move
so total max ways a bug can move w/o coming back to vertice already crossed is 3*2*3 ; 18

Hi Archit3110 ,

To maximize the number of vertex visited, when the bug is on 3rd vertex, it does have 3 possible options but if it chooses to go back to vertex from which it came then the number won't be maximized. so, shouldn't it be 3*2*2? Bunuel
Where am i going wrong?
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Intern  Joined: 14 Apr 2019
Posts: 1
Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Hello,

I believe the 18 originated from the initial possibilities of movement. The bug can go in “3” directions and the probability of any future move is defined by “3!” Thus resulting in 3*3! Which is 3*3*2*1=18.

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Manager  B
Joined: 05 Oct 2017
Posts: 79
Re: A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Hi pushpitkc,
could you help me with the solution of this problem?

TIA
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Manager  G
Joined: 31 Jan 2019
Posts: 164
Location: Switzerland
Concentration: General Management
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A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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Hi,
Hi tried to solve the problem assigning to each step a probability.
as for my solution the probability is: (3/3)*(2/3)*(2/3)*(2/3)*(1/3)*(1/3)*(1/3)=8/729
Can someone explain to why my reasoning is wrong?
Intern  B
Joined: 23 Apr 2019
Posts: 3
Location: India
Schools: ISB '20
GMAT 1: 660 Q49 V31 GPA: 4
A bug starts at one vertex of a cube and moves along the edges of the  [#permalink]

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the bug can move from its arbitrary starting vertex to a neighboring vertex in 3 ways. After this, the bug can move to a new neighbor in 2 ways (it cannot return to the first vertex). The total number of paths is 3^7/2187. Therefore, the probability of the bug following a good path is equal to 6x/2187 for some positive integer x. The only answer choice which can be expressed in this form is 2/243. A bug starts at one vertex of a cube and moves along the edges of the   [#permalink] 02 Jun 2019, 14:08
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