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# What is the maximum number of 3x3 squares that can be formed

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Joined: 09 Feb 2012
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What is the maximum number of 3x3 squares that can be formed  [#permalink]

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Updated on: 06 Jul 2019, 05:18
5
00:00

Difficulty:

45% (medium)

Question Stats:

58% (01:28) correct 42% (01:20) wrong based on 88 sessions

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What is the maximum number of 3x3 squares that can be formed from the squares in a 6x6 checker board?
Attachment:

square.jpg [ 30.23 KiB | Viewed 10322 times ]

A. 4
B. 6
C. 12
D. 16
E. 24

Source: Nova GMAT
Difficulty Level: 650

Originally posted by NYC5648 on 30 May 2012, 13:33.
Last edited by SajjadAhmad on 06 Jul 2019, 05:18, edited 1 time in total.
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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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30 May 2012, 14:14
1
say the board is as below (6*6):

1 2 3 4 5 6
2
3
4
5
6

for 3*3: start from 1st row and 3rd row. Using each row we can have maximum 4, 3*3 square.
we can continue the process for (2nd and 4th row) = 4 squares, (3rd and 5th row) = 4 squares, and (4th and 6th row) = 4 squares
Total = 4+4+4+4=16 (3*3)squares
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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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31 May 2012, 04:48

x x x x x x
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they can be chosen as 4*4 = 16

Top Row: total 4 ways to chose in rows

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1st coloum: Total 4 ways to chose in coloums

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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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31 May 2012, 10:47
Thanks a lot guys. Got it!
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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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28 Jul 2012, 03:43
Is there any direct formula to calculate it?

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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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28 Jul 2012, 04:05
mneeti wrote:
Is there any direct formula to calculate it?

Not really, but no need for a formula.

Think of, for example, choosing your 3 x 3 square by choosing first its left bottom corner. You have to be sure that the upper right corner will still be inside the big board.
It means that you can choose the bottom left corner anywhere in the bottom left area of 4 x 4. Which means 4 x 4 = 16 possibilities.
Or you can concentrate on any other corner of the 3 x 3 square, and then figure out where that corner can be.
I don't think you need to force here some formula saying 4C1 x 4C1 for choosing those corners...or anything similar.

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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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28 Jul 2012, 07:29
I don't like the structure of this question. The question does not explicitly state how large each square is. I know there is a picture, but the question does not reference the picture at all.
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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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26 Aug 2012, 17:16
+1 D

Imagine that tou have a 3x3 square and how you could placed it on that board.
You have to respect the edges or borders
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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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12 Sep 2014, 01:25
1
Answer = 4 * 4 = 16

Refer diagram below:
Attachment:

square.jpg [ 52.28 KiB | Viewed 7839 times ]

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Re: What is the maximum number of 3x3 squares that can be formed  [#permalink]

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07 Mar 2018, 05:47
mneeti wrote:
Is there any direct formula to calculate it?

There is, in fact.

What is the number of k x k squares that can be formed from a n x n board?

(n-k+1)^2, so in that case (6-3+1)^2=16.

The logic is as follows:
-horizontally, you are going to need k boxes of the board for your little k x k square, so n-k boxes will still be "free" (out of the kxk square). Hence, you need to count the number of ways you can allocate those "free" boxes to the left and to the right of your kxk square. You could:
- put 0 box on the left and all the free n-k boxes on the right
- put 1 box on the left and all the free n-k-1 boxes on the right
- ......
- put n-k boxes on the left and 0 box on the right

You can see that there are n-k+1 ways of doing so.

-vertically, the same logic applies.

You therefore end up with the formula above (would also work in higher number of spatial dimensions, eg with cubes, by replacing the ^2 by ^3 in that case.)

Of course, it's better to be able to think about this process and derive the logic during the exam than to use a pre-made formula, but I guess the exercise of deriving a general formula to add layers of complexity is always a fun and rewarding exercise
Re: What is the maximum number of 3x3 squares that can be formed   [#permalink] 07 Mar 2018, 05:47
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