The positive integers from 1 up are arranged, as shown in the table.

Question

Q1. (Algebra) PS - Algebra.jpg The positive integers from 1 up are arranged, as shown in the table above. Nine numbers in the form of a square are chosen from the arrangement. x is the upper-left most number in the square, and the sum of the 9 numbers is 171. What is the value of x? A. 10 B. 11 C. 18 D. 19 E. 23

logro · Accepted Answer

So we pick 9 numbers that are a part of the 3X3 square and add them up

Let us say we pick R1 {4,5,6}, R2 {11,12,13} and R3 {18,19,20} or R1 {3,4,5}, R2 {10,11,12} and R3 {17,18,19) (R1,R2,R3 are the rows in the square}

What we can observe is that within each row the numbers differ by 1. Let x be the left most one of the numbers in the Row R1 and our R1 will be {x, x+1, x+2}

Similarly if we observe that the difference between R1 and R2 (11-4 = 7 or 10-3=7) is 7 , so each rows differ by 7

Which implies that my R2 will be {x+7, x+8, x+9}

similarly R3 {x+14, x+15, x+16}

Adding all the constituents together we get

x+x+1+x+2+x+7+x+8+x+9+x+14+x+15+x+16 = 171

9x+72=171

9x=99

x=11

hence the OA is B

Kudos if it helps!

MathRevolution · Answer

Solution:

Square form:  An Equal number of rows and columns make a square. To form a square of 9 numbers:  3 by 3 square  has to be selected.

When we select the first row, it will have 3 numbers as per the table. So, 
  We will select the 1st, 2nd, and 3rd numbers.
 The next row selection will give us the 8th, 9th, and 10th numbers.
 The third row will give us the 15th, 16th, and 17th numbers.

Given: x is the upper-left most number, so the 1st number is x = 1, and the sum of the 9 numbers is 171.
 
=> 1st number: x
=> 2nd number: x + 1
=> 3rd number: x + 2 ….. and so on.

So, we get x + (x + 1) + (x + 2) + (x + 7) + (x + 8) + (x + 9) + (x + 14) + (x + 15) + (x + 16) = 171.

=> 9x + 72 = 171
=> 9x = 99
=> x = 11

Therefore, B is the correct answer.

Answer: B

MegB07 · Answer

how did we figure that we are taking consecutive numbers and not random selection?

Bunuel · Answer

Because we are taking them in a 3*3 block. That means each row of the square contains three consecutive integers. https://gmatclub.com/forum/download/file.php?id=85561 GMAT-Club-Forum-ji3sef1d.png

AdarshSambare · Answer

This logic worked, but I’m not fully sure why.

Consider the sequence:

```
1 2 3 
4 5 6 
7 8 9
```

* The total sum = **45**, which can also be written as **5 × 9**.
* The average (middle term) = **5**.
* The total number of terms = **9**.

In the problem, we are given:

171 / 9 = 19

So, the average (middle number) is **19**.
Looking at the figure and following the same pattern, we select **11** as the answer.

Answer choice B

luisdicampo · Answer

Deconstructing the Question

The numbers increase by  1  when moving right and by  7  when moving down.

If the upper-left corner of the  3	imes3  square is  x , then all 9 numbers in the square are determined.

Step-by-step

The  3	imes3  square is

x,\ x+1,\ x+2

x+7,\ x+8,\ x+9

x+14,\ x+15,\ x+16

Their sum is  171 , so

x+(x+1)+(x+2)+(x+7)+(x+8)+(x+9)+(x+14)+(x+15)+(x+16)=171

Combine like terms:

9x+72=171

So

9x=99

x=11

A faster way is to use the center of the square.

The center is

x+8

In a symmetric  3	imes3  block, the sum of all 9 numbers equals  9  times the center.

So

171=9(x+8)

x+8=19

x=11

Answer: B

pappal · Answer

x 
 x+1 
 x+2 
 
  x+7 
 x+8 
 x+9 
 
  x+14 
 x+15 
 x+16

whatever square of nine numbers is chosen will follow the above matrix as every no. just below is of +7 form
so 9x+72=171
9x=99
x=11
B

The positive integers from 1 up are arranged, as shown in the table.

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The positive integers from 1 up are arranged, as shown in the table.

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