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# When the figure above is cut along the solid lines, folded along the

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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
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vikasbansal227 wrote:
When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids each with a square base that they share. What is the sum of number of edges and number of faces of this geometric solid?

A. 10
B. 18
C. 20
D. 24
E. 25

Dear vikasbansal227
I'm happy to respond. This is a great question.

First of all, here is a labeled version of the polyhedral net you provided:
Attachment:

octahedron net.JPG [ 21.01 KiB | Viewed 71211 times ]

Incidentally, in your version, you indicate a solid line from B to I, but this should be a dashed line.

When this is folded up, the name of the resultant shape is the Octahedron, one of the five Platonic Solids. It good practice for 3D thinking to study the five Platonic Solids. But, suppose you didn't know about the Octahedron: how would we answer the question?

Notice, that when we fold up, A & C come together, so we could say that B is the vertex of one pyramid, with four edges going down to D, I, J, and A/C. These latter four points form a square that is the "equator" of the shape, so there are four edges around this square. Then, we fold F & H together, forming the second pyramid. This second pyramid has G as its vertex, with four edges come down to the points E, D, I, and F/H; these latter four points also form a square. Then, segment DI acts as a hinge, and along this hinge we fold one pyramid down to meet the other, so that the square bases of the two pyramids meet and become one. Point A/C joins with point E, and point J joins with point F/H.
The final shape has a lower downward-pointing pyramid, with point B as the lower vertex. Four edges go up from this vertex to the square "equator." This square "equator" consists of points D + I + A/C/E + F/H/J, and like any square, it contains four edge segments. Finally, the upper upward-pointing pyramid has vertex G, and four edges go from this vertex down to the four vertices of the square "equator." That's a total of 12 edges.

We can simply count the 8 faces while it is still flat.

E + F = 12 + 8 = 20 ==> OA = (C)

The thing that is a little mind-blowing about the octahedron is this. In the above description, I was discussing the "upper vertex," the "lower vertex," and the "square equator," but because the shape is 100% symmetrical, if we simply turn the shape, any vertex can be the tip of the upper pyramid. There are actually three interlocking squares formed by different combinations of the vertices. This Wikipedia page has a gif of a rotating octahedron, which may help you visualize it more.

Does all this make sense?
Mike
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
mikemcgarry wrote:
vikasbansal227 wrote:
When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids each with a square base that they share. What is the sum of number of edges and number of faces of this geometric solid?

A. 10
B. 18
C. 20
D. 24
E. 25

Dear vikasbansal227
I'm happy to respond. This is a great question.

First of all, here is a labeled version of the polyhedral net you provided:
Attachment:
octahedron net.JPG

Incidentally, in your version, you indicate a solid line from B to I, but this should be a dashed line.

When this is folded up, the name of the resultant shape is the Octahedron, one of the five Platonic Solids. It good practice for 3D thinking to study the five Platonic Solids. But, suppose you didn't know about the Octahedron: how would we answer the question?

Notice, that when we fold up, A & C come together, so we could say that B is the vertex of one pyramid, with four edges going down to D, I, J, and A/C. These latter four points form a square that is the "equator" of the shape, so there are four edges around this square. Then, we fold F & H together, forming the second pyramid. This second pyramid has G as its vertex, with four edges come down to the points E, D, I, and F/H; these latter four points also form a square. Then, segment DI acts as a hinge, and along this hinge we fold one pyramid down to meet the other, so that the square bases of the two pyramids meet and become one. Point A/C joins with point E, and point J joins with point F/H.
The final shape has a lower downward-pointing pyramid, with point B as the lower vertex. Four edges go up from this vertex to the square "equator." This square "equator" consists of points D + I + A/C/E + F/H/J, and like any square, it contains four edge segments. Finally, the upper upward-pointing pyramid has vertex G, and four edges go from this vertex down to the four vertices of the square "equator." That's a total of 12 edges.

We can simply count the 8 faces while it is still flat.

E + F = 12 + 8 = 20 ==> OA = (C)

The thing that is a little mind-blowing about the octahedron is this. In the above description, I was discussing the "upper vertex," the "lower vertex," and the "square equator," but because the shape is 100% symmetrical, if we simply turn the shape, any vertex can be the tip of the upper pyramid. There are actually three interlocking squares formed by different combinations of the vertices. This Wikipedia page has a gif of a rotating octahedron, which may help you visualize it more.

Does all this make sense?
Mike

Hello Mike,
I went through wikipedia link you have mentioned and got to know the shape of octahedron
But i could not interpret the shape from original diagram given in the question
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
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divya517 wrote:
Hello Mike,
I went through wikipedia link you have mentioned and got to know the shape of octahedron
But i could not interpret the shape from original diagram given in the question

Dear divya517,
My friend, this is a HARD question. If the student has run across the octahedron ahead of time, then that probably would give an advantage on this question. The octahedron is a very hard shape to understand, and getting it just from the diagram and the brief verbal description is extremely challenging. If you find 3D Geometry particularly challenging, it might be worthwhile to investigate this a little further to stretch yourself. The octahedron is one of a set of 3D figures known as the "Platonic Solids." The cube is another, and there are five altogether. Investigate these, as well as prisms and pyramids in general. Learning to think in 3D about abstract geometry is a skill that deepen other categories of mathematical thinking.
Does all this make sense?
Mike
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
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mikemcgarry wrote:
divya517 wrote:
Hello Mike,
I went through wikipedia link you have mentioned and got to know the shape of octahedron
But i could not interpret the shape from original diagram given in the question

Dear divya517,
My friend, this is a HARD question. If the student has run across the octahedron ahead of time, then that probably would give an advantage on this question. The octahedron is a very hard shape to understand, and getting it just from the diagram and the brief verbal description is extremely challenging. If you find 3D Geometry particularly challenging, it might be worthwhile to investigate this a little further to stretch yourself. The octahedron is one of a set of 3D figures known as the "Platonic Solids." The cube is another, and there are five altogether. Investigate these, as well as prisms and pyramids in general. Learning to think in 3D about abstract geometry is a skill that deepen other categories of mathematical thinking.
Does all this make sense?
Mike

Hey Mike, I have a general question for problems like this. If I see a figure like this and it's extremely complex to visualize or understand the folds, will the question stem usually tell you the end shape after all the folding? I'm no GMAT expert by any means, but I solved this in under a minute because I basically looked at the figure and thought "to heck with that, can't understand what it's telling me", so I kept reading and the question actually tells me what the final shape is: Two pyramids that share the square face. There's only one way for two pyramids to share the same square face. Everything else was easy, didn't use the figure at all. So again my question to you is, will my method work for other problems similar to this? I want to generalize this so that I can put this in my "pattern recognition" notebook.
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
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Hey Mike, I have a general question for problems like this. If I see a figure like this and it's extremely complex to visualize or understand the folds, will the question stem usually tell you the end shape after all the folding? I'm no GMAT expert by any means, but I solved this in under a minute because I basically looked at the figure and thought "to heck with that, can't understand what it's telling me", so I kept reading and the question actually tells me what the final shape is: Two pyramids that share the square face. There's only one way for two pyramids to share the same square face. Everything else was easy, didn't use the figure at all. So again my question to you is, will my method work for other problems similar to this? I want to generalize this so that I can put this in my "pattern recognition" notebook.

I'm happy to respond. May the Force be with you, my friend.

You know, someone could sit for 100 days in a row, taking a GMAT each day, and might well never see a question anything like this. This is an exceedingly rare topic. This entire discussion concerns something that might about as likely as seeing a unicorn crossing a rainbow!

As a general rule, even hard GMAT math questions are designed to be solved with a certain amount of elegance, and certainly thinking about the figure in this problem just as the two pyramids joined at the bases is a kind of elegant solution, so in the exceedingly rare case that such a question reappeared, I would consider it likely that the text would include some kind of description of the final shape. That way, most folks who have the intelligence to excel in B-school and in the business world could answer the question.

Remember that even with a hard question, the GMAT is trying to discriminate between high scoring students and low scoring students, those for whom B-school will be easy vs. those for whom it will be a challenge. If the test writers just gave the polygonal net with no description of the output shape, then really only folks who had a very strong background in math would get it correct, and 99% of the test-takers would get it wrong. That's not the kind of discrimination that the test is trying to do, separating math geniuses from everyone else. From the perspective of the psychodynamics of testing, such a question would be an abysmal failure, because it simply doesn't select for what the test maker really wants to know.

Remember that the test writers have definite and specific goals in creating this test. The Quant section is a section to select those who will excel in B-school and in the business world. It is not designed to evaluate people pursing doctoral work in Mathematics. Part of being successful on the GMAT is appreciating the test makers' priorities.

Does all this make sense?
Mike
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This link will help to clear any doubt,if any. https://www.dkfindout.com/uk/maths/geome ... d-pyramid/
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VeritasPrepKarishma wrote:

The two pyramids share the square. So the square is hidden inside them. Look at this figure.
Attachment:
The attachment platonic027.gif is no longer available

The colourful figure inside shows two pyramids joined together.

Quote:
"So in all, the figure would have 8 faces (4 triangles + 4 triangles) and 12 edges ( 4 edges where the top triangles join, 4 edges where the bottom triangles join and 4 edges of the square). "
I'm able to understand 8 faces part. But unable to get 12 edges [4edges of top triangle + 4edges of bottom triangle + 4 edges of the square]. When you consider 4 edges of the top triangle that forms the tip of the pyramid, why should it be considered as 4 separate edges. If so all 3 edges of the triangles should also be considered separately, making 12 edges for top triangle and 12 edges for bottom triangle + 4 edges of the square.

When 4 triangles are joined together to make the top figure, we get 4 edges.
Attachment:

image002.gif [ 1.39 KiB | Viewed 63591 times ]
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Does anyone else only count 6 edges? I know 14 is not an answer choice, but counting 12 edges seems to double count in my mind? Like the base of the two pyramids there are 4 edges that both pyramids share, then each of the pyramids has a top edge? Why do we count the base edges twice?
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
brandon7 wrote:
Does anyone else only count 6 edges? I know 14 is not an answer choice, but counting 12 edges seems to double count in my mind? Like the base of the two pyramids there are 4 edges that both pyramids share, then each of the pyramids has a top edge? Why do we count the base edges twice?

What do you mean "top edge"?

The pyramid at the top has 4 edges and the one at the bottom has 4 edges. Looks at the diagram here:
https://gmatclub.com/forum/when-the-fig ... l#p1787756
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brandon7 wrote:
Does anyone else only count 6 edges? I know 14 is not an answer choice, but counting 12 edges seems to double count in my mind? Like the base of the two pyramids there are 4 edges that both pyramids share, then each of the pyramids has a top edge? Why do we count the base edges twice?

You are confusing 'edges' with 'vertices'.

Face: is the flat surface of a solid figure. A face of a solid figure can be a square, a rectangle, a triangle or a circle.
Edge: is the line segment two faces of a solid meet.
Corner (vertex): is a corner where three or more edges of a solid meet.
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
divya517 wrote:
But i could not interpret the shape from original diagram given in the question

You can ignore the diagram completely (as I did when I solved it) because the question tells you that you get a figure with "2 pyramids each with a square base that they share". Imagine what one pyramid looks like - a square with triangles attached to each side and the tips of the triangles stuck together. Imagine how would you have two pyramids with a common base. There would be 4 triangles on the lower side of the square too and their tips would be stuck together. So in all, the figure would have 8 faces (4 triangles + 4 triangles) and 12 edges ( 4 edges where the top triangles join, 4 edges where the bottom triangles join and 4 edges of the square).
All in all you will have 8 + 12 = 20 faces + edges

Why is the face of the square base not being counted? we can also assume the pyramid to be on the same side of the square base and hence 1 should be added to its face.
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
shanks2020 wrote:
divya517 wrote:
But i could not interpret the shape from original diagram given in the question

You can ignore the diagram completely (as I did when I solved it) because the question tells you that you get a figure with "2 pyramids each with a square base that they share". Imagine what one pyramid looks like - a square with triangles attached to each side and the tips of the triangles stuck together. Imagine how would you have two pyramids with a common base. There would be 4 triangles on the lower side of the square too and their tips would be stuck together. So in all, the figure would have 8 faces (4 triangles + 4 triangles) and 12 edges ( 4 edges where the top triangles join, 4 edges where the bottom triangles join and 4 edges of the square).
All in all you will have 8 + 12 = 20 faces + edges

Why is the face of the square base not being counted? we can also assume the pyramid to be on the same side of the square base and hence 1 should be added to its face.

Since they share the base, the pyramids have to be on the opposite sides of the square and will hide the square. If they were on the same side, we will have overlapping triangles and hence get only one pyramid.

Check this diagram again: https://gmatclub.com/forum/when-the-fig ... l#p1787756
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
shanks2020 wrote:
divya517 wrote:
But i could not interpret the shape from original diagram given in the question

You can ignore the diagram completely (as I did when I solved it) because the question tells you that you get a figure with "2 pyramids each with a square base that they share". Imagine what one pyramid looks like - a square with triangles attached to each side and the tips of the triangles stuck together. Imagine how would you have two pyramids with a common base. There would be 4 triangles on the lower side of the square too and their tips would be stuck together. So in all, the figure would have 8 faces (4 triangles + 4 triangles) and 12 edges ( 4 edges where the top triangles join, 4 edges where the bottom triangles join and 4 edges of the square).
All in all you will have 8 + 12 = 20 faces + edges

Why is the face of the square base not being counted? we can also assume the pyramid to be on the same side of the square base and hence 1 should be added to its face.

Since they share the base, the pyramids have to be on the opposite sides of the square and will hide the square. If they were on the same side, we will have overlapping triangles and hence get only one pyramid.

Check this diagram again: https://gmatclub.com/forum/when-the-fig ... l#p1787756

I get your point and understood the diagram. But the question does not state that the sides are equal. Hence, what i felt there is still a possibility of pyramid of different edgle lenght on the same side of square base. Plz. let me know what did i miss ?
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
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shanks2020 wrote:
shanks2020 wrote:
divya517 wrote:
But i could not interpret the shape from original diagram given in the question

You can ignore the diagram completely (as I did when I solved it) because the question tells you that you get a figure with "2 pyramids each with a square base that they share". Imagine what one pyramid looks like - a square with triangles attached to each side and the tips of the triangles stuck together. Imagine how would you have two pyramids with a common base. There would be 4 triangles on the lower side of the square too and their tips would be stuck together. So in all, the figure would have 8 faces (4 triangles + 4 triangles) and 12 edges ( 4 edges where the top triangles join, 4 edges where the bottom triangles join and 4 edges of the square).
All in all you will have 8 + 12 = 20 faces + edges

Why is the face of the square base not being counted? we can also assume the pyramid to be on the same side of the square base and hence 1 should be added to its face.

Since they share the base, the pyramids have to be on the opposite sides of the square and will hide the square. If they were on the same side, we will have overlapping triangles and hence get only one pyramid.

Check this diagram again: https://gmatclub.com/forum/when-the-fig ... l#p1787756

I get your point and understood the diagram. But the question does not state that the sides are equal. Hence, what i felt there is still a possibility of pyramid of different edgle lenght on the same side of square base. Plz. let me know what did i miss ?

You are over complicating it. You know that there is a square in the middle that they share so all 8 triangles have equal bases. When you are folding, how will you get smaller edges? Hence, as per the question it necessarily means the pyramids are on opposite sides.
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
WillGetIt wrote:

When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids each with a square base that they share. What is the sum of number of edges and number of faces of this geometric solid?

A. 10
B. 18
C. 20
D. 24
E. 25

Attachment:
Untitled.png

Approach 1: Just draw the figure

This would be two pyramids essentially glued together at their bases we would have 4*2 = 8 faces, and 4+2(4) edges 4 for the rectangular base, and 2(4) to create the top and bottom four faces

therefore E +F = 8+12 = 20

Approach 2

F-E+V=2 This equality holds for any shape. The problem with this approach is you have to either calculate how many of the folded lines get double counted or just redraw the shape, but if you knew that E=12 and knew we had 6 vertices we can get F=8, therefore E+F=12+8=20. I wouldn't recommend taking this route, as you have to end up redrawing the figure, or pretty much using info you already have, so go with approach 1
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
VeritasPrepKarishma wrote:

The two pyramids share the square. So the square is hidden inside them. Look at this figure.
Attachment:
platonic027.gif

The colourful figure inside shows two pyramids joined together.

Quote:
"So in all, the figure would have 8 faces (4 triangles + 4 triangles) and 12 edges ( 4 edges where the top triangles join, 4 edges where the bottom triangles join and 4 edges of the square). "
I'm able to understand 8 faces part. But unable to get 12 edges [4edges of top triangle + 4edges of bottom triangle + 4 edges of the square]. When you consider 4 edges of the top triangle that forms the tip of the pyramid, why should it be considered as 4 separate edges. If so all 3 edges of the triangles should also be considered separately, making 12 edges for top triangle and 12 edges for bottom triangle + 4 edges of the square.

When 4 triangles are joined together to make the top figure, we get 4 edges.
Attachment:
image002.gif

Hi Karishma, When two triangles are attached to a common base square then should not we just have 4+2 edges ? 4 for the edges which each triangle and square share and two top edges ?
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Re: When the figure above is cut along the solid lines, folded along the [#permalink]
i'm having a hard time seeing how folding makes an Octahedron - i wonder if there is a video illustrating this
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