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If the 4 x 4 grid in the attached picture is filled with the

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If the 4 x 4 grid in the attached picture is filled with the [#permalink] New post 31 Jan 2012, 18:25
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Question Stats:

84% (03:37) correct 15% (10:57) wrong based on 19 sessions
If the 4 x 4 grid in the attached picture is filled with the consecutive integers from 37 to 52, inclusive, so that every row, column and major diagonal sums to the same value, which of the following is a possible value of the sum of the four center cells of the grid (indicated by the shaded area)?

(A) 124
(B) 153
(C) 178
(D) 192
(E) 214

Any idea how to approach this question please?
[Reveal] Spoiler: OA

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Re: Shaded part of the grid [#permalink] New post 01 Feb 2012, 02:58
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Sum of the numbers from 37 to 52 = (n/2)[2a + (n-1)d] = (16/2) [74+15] = 8 * 89 = 712

Therefore the sum of all numbers in the grid = 712
If x is the sum of all numbers in a row/column/major diagonal, then
4x = 712
=> x = 178 = sum of all numbers in any row = sum of all numbers in any diagonal = sum of all numbers in either major diagonal

Now, consider the grid as follows:
1' 2' 3' 4'
5' 6' 7' 8'
9' 10' 11' 12'
13' 14' 15' 16'

We know that 1' + 6' + 11' + 16' = 178
4' + 7' + 10' + 13' = 178
=> 1' + 6' + 11' + 16' + 4' + 7' + 10' + 13' = 356
=> 6' + 7' + 10' + 11' + 1' + 13' + 4' + 16' = 356

Also 5' + 6' + 7' + 8' + 9' + 10' + 11' + 12' = 356
and 1' + 5' + 9' + 13' + 4' + 8' + 12' + 16' = 356
=> 6' + 7' + 10' + 11' - 1' - 13' - 4' - 16' = 0

Therefore 6' + 7' + 10' + 11' = 356/2 = 178 = sum of the middle four numbers

Option (C).
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Re: If the 4 x 4 grid in the attached picture is filled with the [#permalink] New post 15 Nov 2013, 21:23
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Re: If the 4 x 4 grid in the attached picture is filled with the [#permalink] New post 08 Dec 2013, 07:01
In the last step "=> 6' + 7' + 10' + 11' - 1' - 13' - 4' - 16' = 0"
how can we be sure that the +ve groupings and -ve groupings each add up to 178 or 6' + 7' + 10' + 11' = 178. It could be 180 for 6' + 7' + 10' + 11' and 176 for - 1' - 13' - 4' - 16'. Am i missing something here?
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Re: If the 4 x 4 grid in the attached picture is filled with the [#permalink] New post 08 Dec 2013, 07:02
In the last step "=> 6' + 7' + 10' + 11' - 1' - 13' - 4' - 16' = 0"
how can we be sure that the +ve groupings and -ve groupings each add up to 178 or 6' + 7' + 10' + 11' = 178. It could be 180 for 6' + 7' + 10' + 11' and 176 for - 1' - 13' - 4' - 16'. Am i missing something here?
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Re: If the 4 x 4 grid in the attached picture is filled with the [#permalink] New post 10 Dec 2013, 08:16
This is just another great question.
The difficulty level of this question is high and it didn't strike me how to do this question when I tried it in the first attempt.
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Re: If the 4 x 4 grid in the attached picture is filled with the   [#permalink] 10 Dec 2013, 08:16
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