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For a display, identical cubic boxes are stacked in square [#permalink]
23 Sep 2013, 04:10

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A

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E

Difficulty:

5% (low)

Question Stats:

77% (02:25) correct
23% (01:15) wrong based on 161 sessions

For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of the layer has only 1 box, how many boxes are in display?

Re: For a display, identical cubic boxes are stacked in square [#permalink]
23 Sep 2013, 05:23

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imhimanshu wrote:

For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of the layer has only 1 box, how many boxes are in display?

A. 236 B. 260 C. 269 D. 276 E. 285

Basically we have a 9-layer pyramid as shown below:

Attachment:

pyramid_with_corner_cube_from_istock.jpg [ 9.94 KiB | Viewed 10201 times ]

(Actually this pyramid 8-layer, couldn't find 9-layer one image)

The number of boxes would be: 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1 = 285.

You can use the sum of the first n perfect squares formula to calculate: \(\frac{n(n+1)(2n+1)}{6}=\frac{9*(9+1)(2*9+1)}{6}=285\).

Re: For a display, identical cubic boxes are stacked in square [#permalink]
23 Sep 2013, 13:46

Thanks bunuel but how did you manage to understand that? I read the question again and cannot imagine the picture you uploaded in my head at all. Thanks! _________________

Re: For a display, identical cubic boxes are stacked in square [#permalink]
23 Sep 2013, 23:56

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roygush wrote:

Thanks bunuel but how did you manage to understand that? I read the question again and cannot imagine the picture you uploaded in my head at all. Thanks!

I read the stem carefully. We are told that: Cubic boxes are stacked in square layers --> each layer is a square; The bottom of the layer has 81 boxes --> the bottom layer has 9 rows and each row has 9 boxes. Each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it --> the second layer has 8 rows and each row has 8 boxes. ...

Re: For a display, identical cubic boxes are stacked in square [#permalink]
24 Sep 2013, 07:58

Bunuel wrote:

roygush wrote:

Thanks bunuel but how did you manage to understand that? I read the question again and cannot imagine the picture you uploaded in my head at all. Thanks!

I read the stem carefully. We are told that: Cubic boxes are stacked in square layers --> each layer is a square; The bottom of the layer has 81 boxes --> the bottom layer has 9 rows and each row has 9 boxes. Each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it --> the second layer has 8 rows and each row has 8 boxes. ...

Hope it helps.

I got confused (and I still am) by the line which says "Each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it"

I got that the bottom layer will have 9 x 9 boxes I also understand that the next level up will have 8 rows of boxes and since this layer also has to form a square hence it needs 8 boxes in the column as well.

What is elusive for me is "and 1 fewer box in each remaining row[/color] than the layer directly below it". Can you please explain again.

Re: For a display, identical cubic boxes are stacked in square [#permalink]
24 Sep 2013, 14:04

1

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Expert's post

violetsplash wrote:

Bunuel wrote:

roygush wrote:

Thanks bunuel but how did you manage to understand that? I read the question again and cannot imagine the picture you uploaded in my head at all. Thanks!

I read the stem carefully. We are told that: Cubic boxes are stacked in square layers --> each layer is a square; The bottom of the layer has 81 boxes --> the bottom layer has 9 rows and each row has 9 boxes. Each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it --> the second layer has 8 rows and each row has 8 boxes. ...

Hope it helps.

I got confused (and I still am) by the line which says "Each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it"

I got that the bottom layer will have 9 x 9 boxes I also understand that the next level up will have 8 rows of boxes and since this layer also has to form a square hence it needs 8 boxes in the column as well.

What is elusive for me is "and 1 fewer box in each remaining row[/color] than the layer directly below it". Can you please explain again.

Each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it:

1st layer has 9 rows and 9 boxes in each of them. 2nd row has 1 fewer, so 8 rows and each of the remaining 8 rows has 1 fewer box, so 8 boxes in it.

Re: For a display, identical cubic boxes are stacked in square [#permalink]
04 Apr 2015, 16:42

Bunuel wrote:

imhimanshu wrote:

For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of the layer has only 1 box, how many boxes are in display?

A. 236 B. 260 C. 269 D. 276 E. 285

Basically we have a 9-layer pyramid as shown below:

Attachment:

pyramid_with_corner_cube_from_istock.jpg

(Actually this pyramid 8-layer, couldn't find 9-layer one image)

The number of boxes would be: 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1 = 285.

You can use the sum of the first n perfect squares formula to calculate: \(\frac{n(n+1)(2n+1)}{6}=\frac{9*(9+1)(2*9+1)}{6}=285\).

Answer: E.

Hope it's clear.

Is there any chance you can apply why that 6 is there? I want to make sure I can apply this formula in more complicated cases.

gmatclubot

Re: For a display, identical cubic boxes are stacked in square
[#permalink]
04 Apr 2015, 16:42

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