An ant is clinging to one corner of a box in the shape of a

Make the cube and select two opposite corners. One where the ant is right now and the second is where it wants to reach.
Notice that from its current position, it has 3 different paths that it can take i.e. it has 3 possible options. Make the ant move on any one path out of these 3.
Once one path is chosen, it has two different paths it can take (it cannot take the third one since it cannot revisit the third point). Make the ant move on any one path out of these 2.
Now, from this point, it again has 3 options to reach the point where it wanted to reach. Notice that the case will be the same even if you had selected different paths in the first two moves because it is a cube so all sides are the same.

Total number of different paths = 3*2*3 = 18
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Here is the reason the number of paths is 3*2*3 and how it takes everything into consideration:


I think the first two steps are clear so the first step is taken in 3 ways and second step in 2 ways. Now, draw the cube and see that the ant would be at one of three points (the points where you reach after traversing 2 edges) - E, C, G

For each of these points, there are 3 unique ways in which the ant can reach the desired vertex.

1st way - Directly go to the desired point. If it is an E, this means EF
2nd way - Go via 3 edges e.g. if it came from AD- DE, it goes EH-HG-GF
3rd way - Go via 5 edges e.g. if it came from AD- DE, it goes EH-HG-GB-BC-CF
Since it is a cube, we know that what works for one edge, will work for other two as well.

So you multiply 3*2 by 3 to get 18.
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guys any idea how this problem can be solved algebraically? the OE essentially sketches out the different paths, which will be very tedious on G day.

@ Bunuel I have to type it all over again coz u deleted my last post..

Here how does the ant have 3 choices after the second path..It again has 2 choices..
Also it misses all the complex paths that the ant visits not in 3 steps..say in 5 or 7 steps..(It is possible)

Either I misunderstood the Q and the E or the explanation is incorrect

I guess the way would be:
No of ways to do in-
3 steps: 3*2*1=6
5 steps: 3*2*1*1*1=6
7 steps: 3*2*1*1*1*1*1=6
No more steps are possible: total 18 ways

Experts please comment

Hi Karishma
I could not get how there are three paths from the second point. Could you please explain in detail. Thanks

Bumping for review...Karishma & Bunuel..please review the solution in view of the above approach I have posted

As for your solution, it is correct too, of course. You chose to split it into "number of paths used". But note the 1s you used in 5 steps and 7 steps. It means that paths are unique and the method involves some unnecessary counting. A cube gives us symmetry and it would be good to use that.
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At first stage ant has three ways, right?
At the second stage ant has only two ways, since it cannot return back on its path.
At the third stage Ant has only one way to its destination.
Henceforth, 3x2x1=6

Since there are three paths at the first stage we just need to add the outcomes of each three ways:
6+6+6=18

VeritasPrepKarishma ma'am
I still didn't understand the 3rd multiple of 3. Could you please explain it again?
Considering the ant starts from E and its destination being B, From point E it has 3 distinct path.
When it gets to point D,H or F it has 2 distinct path through which it can reach B.
Now comes the third point. From D it can come to A or C, From H it can go to A or G and from F it can go to C or G again giving rise to 2 distinct path.
Ma'am could you please explain from here?

I have discussed this question here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html/2015/0 ... -the-gmat/

See if it helps. Let me know if something is still unclear.
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This question has a bit of IQ test feel to it. You need to be able to visualize the paths to fit the problem to a standard permutations/combinations calculation.

Sarah

Let’s say the ant is currently at corner A, thus it wants to go to corner G since it’s the furthest from A.

Now from A, its next move is to B, to E, or to D. Let’s analyze the number of paths it can have if its next move is to B:

A-B-F-G

A-B-F-E-H-G

A-B-F-E-H-D-C-G

A-B-C-G

A-B-C-D-H-G

A-B-C-D-H-E-F-G

We see that we have 6 paths if its first move is to B. Without listing the actual paths, if its first move is to E, there should be also 6 paths, and another 6 paths if its first move is to D. Therefore, there will be a total of 6 + 6 + 6 = 18 different paths from A to G.

Answer: C
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we have 12 edges and 6 faces in a cube, the ant got 12+6=18 ways

answer is C

Ant has to add 1 unit in x, y and z direction in order to reach to distant corner

Case 1 (+1, +1, +1)= 3*2*1=6
Case 2 (+1, +1, -1, +1, +1)= 3*2*1*1*1=6
Case 3 (+1, +1, -1, +1, -1, +1, +1)= 3*2*1*1*1*1*1=6

Total number of ways = 6+6+6=18

Different ways the ant can move is,
1. 3-edges path(1-1-1)
2. 5-edges path(1-1-3)
3. 7-edges path(1-1-5)

Now, assuming that the ant doesn't feel adventurous and ventures back on the same path it crawled upon previously,
Number of ways from Start Point (1) = 3(1 OR 1 OR 1)
AND
Number of ways from Mid Point (2) = 2(1 OR 1)
AND
Number of ways from Last Point (3) = 3(1 OR 1 OR 1)

Total ways = 3x2x3 = 18

On each face of the cube, the ant can crawl on three sides (except the one that leads to its starting point) and there are six faces for the cube, so the total number of ways for the ant to reach to the end is 6*3 = 18 ways

VeritasKarishma Mam I saw your cube diagram for the explanation. Though 1st two steps are easy to understand. But when it reaches E, C or G why won't it straightaway go to point F ?

For instance in this path 2nd way - Go via 3 edges e.g. if it came from AD- DE, it goes EH-HG-GF, when there is a way from point E to point F why go down,then right and then up to reach the point F ? So according to me it should be 3*2*1=6 ways.

Other experts please help Bunuel chetan2u ManhattanPrep

The question is "in how many different ways can it go to the distant corner?" Hence, we need to consider ALL ways, even if some of them are not the most efficient. Yes, it can go directly to F and that is one way to reach it. But it can also take a longer way and it will still reach F and that is also another way to reach F. You need to consider ALL different ways.
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