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For a display, identical cubic boxes are stacked in square [#permalink]

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23 Sep 2013, 05:10

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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?

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 29564 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]

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23 Sep 2013, 14:46

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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!
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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]

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24 Sep 2013, 08:58

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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.

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]

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04 Apr 2015, 17:42

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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]

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11 Oct 2015, 19:42

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]

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03 Jan 2016, 10:17

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.

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

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29 Aug 2016, 13:01

Attachment:

File comment: Visualize the Question this Way

IMG_20160830_012335.jpg [ 1.04 MiB | Viewed 7919 times ]

In the above figures ,small circles are akin to cubic boxes and lines are rows. So bottom most layer has 9*9 = 81 boxes . Now this figure has more layers stacked on top of it and each layer has 1 less box(small circle) and 1 less row( line). If you follow this theory then you will notice that no of rows = no of boxes in each row.

So 2nd layer from the top will have 2 rows with 2 boxes each. Similarly top most layer will have 1 row and 1 box.

So the total no of boxes will be : 9* 9 + 8*8 +.....+ 1*1 = 285 .

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 layer has 21 boxes and the top layer has only 1 box, how many boxes are in the display?

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

Hi, The Q seems to be flawed as 21 cannot be the number of boxes . It is either 81 or 25.

You can imagine this as a huge cube from which steps are made in 2 sides by removing a row of boxes in each layer. In numerical value,all the layers will be square of integers starting from 1 on top to 5 or 9 in lowermost.

When we add these, it becomes \(1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+9^2\) Either add all of them or use formula SUM=\(\frac{n(n+1)(2n+1)}{6}\)=9*(9+1)*(2*9+1)/6=9*10*19/6=15*19=285 E
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