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

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What is the maximum number of 3x3 squares that can be formed [#permalink] New post   25 Mar 2019, 05:07
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What is the maximum number of 3x3 squares that can be formed from the squares in the 6x6 checkerboard to the right?

(A) 4

(B) 6

(C) 12

(D) 16

(E) 24
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D
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sal60
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink] New post   23 Nov 2019, 13:22
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Let's consider the picture above.
For each row we can pick exactly 4 different groups of 3 consecutive boxes.
In the same manner, for each column we can select 4 different groups of 3 consecutive boxes.
Therefore, snce we are required the maximum number of 3x3 squares of the chessboard,
we need to combine our combinations:
4 * 4 = 16
Answer is D.
Hope is clear.
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mouda
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink] New post   23 Nov 2019, 13:57
sal60 wrote:
Let's consider the picture above.
For each row we can pick exactly 4 different groups of 3 consecutive boxes.
In the same manner, for each column we can select 4 different groups of 3 consecutive boxes.
Therefore, snce we are required the maximum number of 3x3 squares of the chessboard,
we need to combine our combinations:
4 * 4 = 16
Answer is D.
Hope is clear.



Can you elaborate a little bit. I do not know how you get the 4*4
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sal60
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink] New post   04 Dec 2019, 12:46
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I believe an image may better explain it:

"For each row we can pick exactly 4 different groups of 3 consecutive boxes.
In the same manner, for each column we can select 4 different groups of 3 consecutive boxes."
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praveensamudrala
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink] New post   05 Dec 2019, 02:02
Grouping the top 3 rows, among the 6 columns, adjacent 3 columns can be selected in 4 ways which can give 3x3 box. Similarly when done for columns, we get 4 ways for 3x3 box. So, total number can be 4*4 = 16 boxes.
My explanation might be not very clear, but thanks to other user who took sometime to explain with images.
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink] New post   14 Aug 2021, 14:57
Is there a way to write this in combinatorics???
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink] New post   14 Aug 2021, 16:31
GonzoG wrote:
Is there a way to write this in combinatorics???

the easiest IMO, it's a simple counting problem 4x4 of three-square figure in the upper left corner
the figure may move 4 times leftward and 4 times downward, in total 16 times
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Re: What is the maximum number of 3x3 squares that can be formed [#permalink]
14 Aug 2021, 16:31
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