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In how many ways can the numbers 1 through9  be placed once each in the squares below, if the numbers 1  and 2 cannot be placed in adjacent squares (either horizontally, vertically, or diagonally), and rotations and reflections of the grid are considered the same?


The squares in the grid are all the same size and it is a 3x3 grid (3 rows and 3 columns)


Only possible way I think of is to fill in the squares with numbers and see if they work sort of a trial and error

 Aug 9, 2022
 #1
avatar+124596 
+1

  ____    _____     _____

     1          2             3

_____   _____    ______

    4          5             6

 _____   _____   ______

   7           8            9

 

If the "1"  is placed in  any of the 4 corner positions

For each of these, the "2"  can  occupy any  5 positions and the other 7 digits can occupy the other 7 positions in 7! ways

This gives us  4  * 5 * 7!  =  100800 ways 

 

If the "1"  is placed in positions "4"  or "6"  the "2"  can only occupy only 3 positions and the other digits can  be arranged in  7! ways  =   2 * 3 * 7!  = 30240 ways

 

If the "1" is placed  in positions  "2" or "8"  the  "2" can only occupy 3 positions and the other 7 digits can  occupy the other 7 positions in 7!  ways =  2 * 3 * 7!  = 30240 ways

 

Finally.....the "1" cannot be placed in  the middle position because the "2" will have to be next to it in some way

 

So....the total ways =   100800 + 2(30240)  = 161280 ways

 

cool cool cool

 Aug 9, 2022
 #2
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I understand the logic but I don't think that would be correct.

 Aug 9, 2022
 #3
avatar+2446 
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I think it has to do with the reflection and rotation part...

                                                                                                                                                                                                                     1 2 3

For example, with Chris's method, he counts for each of these below, which can all be reflected and or rotated to get the original grid (4 5 6) 

                                                                                                                                                                                                                     7 8 9  

 

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

4 5 6    8 5 2    6 5 4    2 5 8

7 8 9    9 6 3    3 2 1    1 4 7

 

3 2 1    9 6 3    7 8 9    9 6 3

6 5 4    8 5 2    4 5 6    8 5 2

9 8 7    7 4 1    1 2 3    7 4 1

 

7 8 9    1 4 7    3 2 1    1 4 7

4 5 6    2 5 8    6 5 4    2 5 8

1 2 3    3 6 9    9 8 7    3 6 9

 

9 8 7    3 6 9    1 2 3    7 4 1

6 5 4    2 5 8    4 5 6    8 5 2

3 2 1    1 4 7    7 8 9    9 6 3

 

There are 4 ways to rotate the square, and 4 ways to reflect it, so i think it is \(161280 \div 16 = \color{brown}\boxed{10080}\) ways

 Aug 9, 2022
edited by BuilderBoi  Aug 9, 2022

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