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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:
78% (02:14) correct
22% (01:22) wrong based on 250 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 16465 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
1
This post received KUDOS
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.
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.
Re: For a display, identical cubic boxes are stacked in square [#permalink]
03 Jan 2016, 09:17
I liked plaverbach approach, as I couldn't understand the question in the first place and choose random wrong answer. But when I saw the picture posted, I could use the plaverbach approach.
gmatclubot
Re: For a display, identical cubic boxes are stacked in square
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03 Jan 2016, 09:17
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