# Math Help - 2 questions 6 digit number and neighbour squares

1. ## 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

2. 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!

3. 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!!

4. 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: $6^2 = 36$ positions.

Hence, the 8 forms can be placed in: . $49\cdot36 \:=\:{\color{blue}1764}$ 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: . $4\cdot35 \:=\:140$ ways.

Hence, the three horizontal forms have: . $3 \times 140 \:=\:420$ 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: . $3 \times 140\:=\:420$ ways.

Therefore, these three forms have: . $420 + 420 \:=\:{\color{blue}840}$ 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 $32 + 32 \:=\:{\color{blue}64}$ ways.

Therefore, there are: . $1764 + 840 + 64 \:=\:\boxed{2668}$ ways
. . to place the twelve pentominos on a chessboard.

I'm grateful they didn't ask for "six neighbour squares".
There would be $35$ 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.
.