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A certain cube is composed of 1000 smaller cubes, arranged 1

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mohnish104
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A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   30 Nov 2013, 05:02
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A certain cube is composed of 1000 smaller cubes, arranged 10 by 10 by 10. The top layer of cubes is removed from a face, then from the adjacent face above it, then from the adjacent face to the right of the first. The process is repeated on the same three faces in reverse order. Finally, a last layer is taken from the first face. How many smaller cubes have been removed from the larger cube?

(A) 488
(B) 552
(C) 612
(D) 722
(E) 900
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B
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Bunuel
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   02 Dec 2013, 02:41
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mohnish104 wrote:
A certain cube is composed of 1000 smaller cubes, arranged 10 by 10 by 10. The top layer of cubes is removed from a face, then from the adjacent face above it, then from the adjacent face to the right of the first. The process is repeated on the same three faces in reverse order. Finally, a last layer is taken from the first face. How many smaller cubes have been removed from the larger cube?

(A) 488
(B) 552
(C) 612
(D) 722
(E) 900


{width}*{length}*{height} = 10*10*10.

The top layer of cubes is removed from a face, we get 10*9*10;
Then from the adjacent face above it, we get 10*9*9
Then from the adjacent face to the right of the first, we get 9*9*9.

The process is repeated on the same three faces in reverse order, we get 8*8*8.

Finally, a last layer is taken from the first face, we get 8*7*8=448.

{# of removed cubes} = {original} - {final} = 1,000 - 448 = 552.

Answer: B.

Hope it's clear.
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   14 Jan 2017, 17:17
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We can answer this by keeping track of how many cubes are lopped off of each side as the cube is trimmed (10 x 10 + 10 x 9 + 9 x 9 + ...), but this approach is tedious and error prone. A more efficient method is to determine the final dimensions of the trimmed cube, then find the difference between the dimensions of the trimmed and original cubes.

Let's call the first face A, second face B, and third face C. By the end of the operation, we will have removed 2 layers each from faces B and C, and 3 layers from face A. So B now is 8 cubes long, C is 8 cubes long, and A is 7 cubes long. The resulting solid has dimensions 8 x 8 x 7 cubes or 448 cubes. We began with 1000 cubes, so 1000 - 448 = 552. Thus, 552 cubes have been removed.

The correct answer is B
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   15 Aug 2016, 06:15
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The cube that remains is 7*8*8 (we took three layers from one side, and two from the other two sides, order does not matter)
(10-3)(10-2)(10-)2=7*8*8=448
1000-448=552

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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   30 Nov 2013, 11:52
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Hint: Once you see what's happening with the first few operations, this might be a good candidate for ballparking.
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   04 Sep 2018, 23:39
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Bunuel wrote:
mohnish104 wrote:
A certain cube is composed of 1000 smaller cubes, arranged 10 by 10 by 10. The top layer of cubes is removed from a face, then from the adjacent face above it, then from the adjacent face to the right of the first. The process is repeated on the same three faces in reverse order. Finally, a last layer is taken from the first face. How many smaller cubes have been removed from the larger cube?

(A) 488
(B) 552
(C) 612
(D) 722
(E) 900


{width}*{length}*{height} = 10*10*10.

The top layer of cubes is removed from a face, we get 10*9*10;
Then from the adjacent face above it, we get 10*9*9
Then from the adjacent face to the right of the first, we get 9*9*9.

The process is repeated on the same three faces in reverse order, we get 8*8*8.

Finally, a last layer is taken from the first face, we get 8*7*8=448.

{# of removed cubes} = {original} - {final} = 1,000 - 448 = 552.

Answer: B.

Hope it's clear.



Understanding this question itself will take around 2 mins, plus 30-40 sec for solving, shall these be attempted or should be guessed and moved over in real test scenarios ?
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   04 Dec 2013, 10:28
Are there any more problems similar to this to practice? I got the steps right but it took me a loooooooong time.

And why does 7x8x8 represent the number of cubes removed? Isn't that the number of cubes remaining of the larger cube?

Never mind, I got it - I misread the question.
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   01 Oct 2016, 23:41
I rushed it, it's definitely easier to do it the way mentioned above. I removed them and slowly added them to one another, wasted time doing so.
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   25 Jun 2020, 02:59
mohnish104 wrote:
The top layer of cubes is removed from a face, then from the adjacent face above it, then from the adjacent face to the right of the first. The process is repeated on the same three faces in reverse order. Finally, a last layer is taken from the first face.

Dear IanStewart,

I completely cannot understand what the question is saying. I'm lost :(
This may not be a great question, but could you please explain how do we know which side that the question is referring to?
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   25 Jun 2020, 05:19
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varotkorn wrote:
I completely cannot understand what the question is saying. I'm lost :(
This may not be a great question, but could you please explain how do we know which side that the question is referring to?


I don't know the source, but it's clearly not an official problem. It's not worded the way an official question would be. For one thing, it assumes an abstract cube has some particular orientation (when it takes about the face "to the right" of another), which is not the case - there are four faces "to the right" of another face depending how you look at a cube. There's no value in expending effort decoding the wording of questions that aren't written the way official questions are written.
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   08 Jul 2021, 05:31
Bunuel wrote:
mohnish104 wrote:
A certain cube is composed of 1000 smaller cubes, arranged 10 by 10 by 10. The top layer of cubes is removed from a face, then from the adjacent face above it, then from the adjacent face to the right of the first. The process is repeated on the same three faces in reverse order. Finally, a last layer is taken from the first face. How many smaller cubes have been removed from the larger cube?

(A) 488
(B) 552
(C) 612
(D) 722
(E) 900


{width}*{length}*{height} = 10*10*10.

The top layer of cubes is removed from a face, we get 10*9*10;
Then from the adjacent face above it, we get 10*9*9
Then from the adjacent face to the right of the first, we get 9*9*9.

The process is repeated on the same three faces in reverse order, we get 8*8*8.

Finally, a last layer is taken from the first face, we get 8*7*8=448.

{# of removed cubes} = {original} - {final} = 1,000 - 448 = 552.

Answer: B.

Hope it's clear.


i got your point 10*10*10.
as width , but every face is a square that means 100=10*10*10.
but if you remove top face its equivalent to removing top square isnt it which is 100??
thus didnt undertsand 10*10*9?
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Re: A certain cube is composed of 1000 smaller cubes, arranged 1 [#permalink] New post   12 Nov 2023, 23:50
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