# 2 questions 6 digit number and neighbour squares

• Mar 27th 2009, 02:19 PM
Innsmouth
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
• Mar 27th 2009, 06:54 PM
Soroban
Hello, Innsmouth!

Quote:

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!

• Mar 28th 2009, 08:54 AM
Quote:

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 (Surprised)

after you solved it ...

Absolutely fabulous!! (Yes)
• Mar 28th 2009, 03:16 PM
Soroban
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: $\displaystyle 6^2 = 36$ positions.

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

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

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

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

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