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

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Question Stats:

77% (02:14) correct
23% (01:20) wrong based on 204 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 13565 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

1

This post received KUDOS

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

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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.

Re: For a display, identical cubic boxes are stacked in square [#permalink]
11 Oct 2015, 18:42

Here's where non-native speakers could have trouble. By display I kept thinking of a computer display and I tried to visualize boxes arranged within the TV, and jumped into the conclusion that this was similar to a problem in the OG (13th Ed. PS 124). Now, if you don't know the formula for the sum of the first n perfect squares (I actually forgot it on a second attempt), it is just nonsense to sum each square result. plaverbach's approach is the appropriate one. After taking a look at the answers and noticing that only two of them have the same units number, you pray that those are wrong and go ahead and find that unit.

gmatclubot

Re: For a display, identical cubic boxes are stacked in square
[#permalink]
11 Oct 2015, 18:42

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