Thread: 2 questions 6 digit number and neighbour squares

Neighbour Squares
(4)On a standard chessboard how many different ways can you select a block of five neighbour squares?

Two squares are neighbours if they have a common side.

(If the problem was asked for a 3x3 board the answer would be 49.)

enjoy it

Hello, Innsmouth!

Neighbour Squares
On a standard chessboard how many different ways
. . can you select a block of five neighbour squares?
Two squares are neighbours if they have a common side.

(If the problem was asked for a 3x3 board the answer would be 49.)

I finally found the 49 ways on a 3x3 board!

Code:

      *-----------*
      |:::::::::::|
      |:::::::*---*
      |:::::::|   |    16 ways
      *---*---*   |
      |           |
      *-----------*

Code:

      *-----------*
      |:::|:::|:::|
      *:::*---*:::*
      |:::|   |:::|    8 ways
      *---*   *---*
      |           |
      *-----------*

Code:

      *-------*---*
      |:::::::|   |
      *---*:::*---*
      |   |:::::::|    8 ways
      |   |:::*---*
      |   |:::|   |
      *---*---*---*

Code:

      *-----------*
      |:::::::::::|
      |:::*-------*
      |:::|       |    4 ways
      |:::|       |
      |:::|       |
      *---*-------*

Code:

      *-----------*
      |:::::::::::|
      *---*:::*---*
      |   |:::|   |    4 ways
      |   |:::|   |
      |   |:::|   |
      *---*---*---*

Code:

      *---*-------*
      |   |:::::::|
      *   |:::*---*
      |   |:::|   |    4 ways
      *---*:::|   *
      |:::::::|   |
      *---*---*---*

Code:

      *---*-------*
      |:::|       |
      |:::*---*   |
      |:::::::|   |    4 ways
      *---*:::*---*
      |   |:::|:::|
      *---*---*---*

Code:

      *---*---*---*
      |   |:::|   |
      *---*:::*---*
      |:::::::::::|    1 way
      *---*:::*---*
      |   |:::|   |
      *---*---*---*

On an 8x8 board, these would have many more placements.

Plus, we must place the other four "Pentominoes".

Code:

      *-------*
      |:::::::|
      *---*:::*-------*
          |:::::::::::|
          *-----------*


      *---*
      |:::|
      |:::*-----------*
      |:::::::::::::::|
      *---------------*


          *---*
          |:::|
      *---*:::*-------*
      |:::::::::::::::|
      *---------------*


      *-------------------*
      |:::::::::::::::::::|
      *-------------------*

That's a LOT of counting!

Originally Posted by Soroban

Hello, Innsmouth!


I finally found the 49 ways on a 3x3 board!

Code:

      *-----------*
      |:::::::::::|
      |:::::::*---*
      |:::::::|   |    16 ways
      *---*---*   |
      |           |
      *-----------*

Code:

      *-----------*
      |:::|:::|:::|
      *:::*---*:::*
      |:::|   |:::|    8 ways
      *---*   *---*
      |           |
      *-----------*

Code:

      *-------*---*
      |:::::::|   |
      *---*:::*---*
      |   |:::::::|    8 ways
      |   |:::*---*
      |   |:::|   |
      *---*---*---*

Code:

      *-----------*
      |:::::::::::|
      |:::*-------*
      |:::|       |    4 ways
      |:::|       |
      |:::|       |
      *---*-------*

Code:

      *-----------*
      |:::::::::::|
      *---*:::*---*
      |   |:::|   |    4 ways
      |   |:::|   |
      |   |:::|   |
      *---*---*---*

Code:

      *---*-------*
      |   |:::::::|
      *   |:::*---*
      |   |:::|   |    4 ways
      *---*:::|   *
      |:::::::|   |
      *---*---*---*

Code:

      *---*-------*
      |:::|       |
      |:::*---*   |
      |:::::::|   |    4 ways
      *---*:::*---*
      |   |:::|:::|
      *---*---*---*

Code:

      *---*---*---*
      |   |:::|   |
      *---*:::*---*
      |:::::::::::|    1 way
      *---*:::*---*
      |   |:::|   |
      *---*---*---*

On an 8x8 board, these would have many more placements.

Plus, we must place the other four "Pentominoes".

Code:

      *-------*
      |:::::::|
      *---*:::*-------*
          |:::::::::::|
          *-----------*


      *---*
      |:::|
      |:::*-----------*
      |:::::::::::::::|
      *---------------*


          *---*
          |:::|
      *---*:::*-------*
      |:::::::::::::::|
      *---------------*


      *-------------------*
      |:::::::::::::::::::|
      *-------------------*

That's a LOT of counting!

If you won't mind me asking , how much time did it take to

post that

after you solved it ...

Absolutely fabulous!!

Hello again, Innsmouth!

I was wrong . . . It didn't take that much work to count them.


The 8 pentominos can be placed in a 3x3 grid in 49 ways.
The 3x3 grid can be located in: positions.

Hence, the 8 forms can be placed in: . ways.



Consider 3 of the remaining 4 pentominoes.

Code:

      *-------*
      |:::::::|
      *---*:::*-------*
          |:::::::::::|
          *-----------*

      *---*
      |:::|
      |:::*-----------*
      |:::::::::::::::|
      *---------------*

          *---*
          |:::|
      *---*:::*-------*
      |:::::::::::::::|
      *---------------*

Horizontal: these occupy a 2x4 rectangle; each has 4 orientations.
The 2x4 rectangles have 5 horizontal locations and 7 vertical locations.
. . The rectangles have 35 possible positions.
So, each form has: . ways.

Hence, the three horizontal forms have: . ways.


Vertical: the forms occupy a 2x4 rectangle, with 4 orientations.
The rectangles have 7 horizontal locations and 5 vertical locations.

Hence, the three vertical forms have: . ways.


Therefore, these three forms have: . ways.

Code:

      *-------------------*
      |:::::::::::::::::::|
      *-------------------*

The horizontal 1x5 bar has 4 horizontal locations and 8 vertical locations.
. . Hence, it has 32 possible locations.
The vertical 5x1 bar also has 32 possible locations.

Hence, it has ways.


Therefore, there are: . ways
. . to place the twelve pentominos on a chessboard.


I'm grateful they didn't ask for "six neighbour squares".
There would be hexominos to consider.


Thanks for the praise, ARDASH. I didn't pay much attention,
but it probably took an hour (most work on diagrams, using
COPY and PASTE over and over).
It's certainly obvious by now, isn't it? . . . I truly enjoy doing this.
.