Don't ask me any questions about it since what you see is what there is.  I'll post the solution next week.  Hope I'm not repeating one that has been posted before. I did a search and could not find it.

 

 

Imagine that we define new chess pieces, with new powers.
Each piece has an associated "hit matrix" with 15 columns and 15 rows.
For example the following matrix defines the traditional rook.

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 R 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

To interpret this matrix: if the rook were at the center (labelled "R"), any piece at a square labelled "1" would be attacked by the rook,
but a piece at a square labelled "0" would not be attacked.
To decide whether a rook attacks a square on an 8 by 8 chessboard, we superimpose the chessboard on this matrix so that the rook is on the "R",
and look at the corresponding square.
In that sense a rook's power is independent of position: if a rook at a4 attacks a piece two squares to the right, then a rook at b7 also attacks a piece two square to the right.

With that in mind, we can define a new chess piece by writing such a 15 by 15 matrix of 0's and 1's (with the piece at the center).
In fact we can define 2^224 different kinds of piece. (Of the 225 entries in the matrix, only the central one is irrelevant.)

For each such chess piece, we define its "value" as follows:
Given a piece of type F, suppose that the largest number of such pieces that can be placed on the 8x8 board without attacking each other, is k.
Then a piece of type F has value 1/k.

Our goal is to define a number of new pieces, and place them on the 8x8 board without attacking each other.
We want to maximize the total value of the pieces on the board.

An example follows:

Define a piece F by the matrix

0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 F 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1

Such a piece F can attack anything in any column to its right, or anything above it on its own column.
Its value is 1, since no two F pieces can be placed on the board without one attacking the other.

Similarly define G by

1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 G 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

G also has value 1.

Then the following 8x8 board would have value 2.75 = (6/8) + (1) + (1). It has one each of the new G and F pieces, along with 6 rooks (value=1/8).

G 0 0 0 0 0 0 0
0 R 0 0 0 0 0 0
0 0 R 0 0 0 0 0
0 0 0 R 0 0 0 0
0 0 0 0 R 0 0 0
0 0 0 0 0 R 0 0
0 0 0 0 0 0 R 0
0 0 0 0 0 0 0 F

To save time and space, this board could be abbreviated as follows:

Value = 2.75

1 0 0 0 0 0 0 0
0 8 0 0 0 0 0 0
0 0 8 0 0 0 0 0
0 0 0 8 0 0 0 0
0 0 0 0 8 0 0 0       - This should be in red
0 0 0 0 0 8 0 0
0 0 0 0 0 0 8 0
0 0 0 0 0 0 0 1

That is, an 8 by 8 array of integers, with 0 denoting an empty square, and a positive integer k denoting a piece with value 1/k (because a maximum of k copies of that particular piece could be placed on the board).

Problem: design a collection of pieces, place them on the board, obtain a value larger than 20, and send to us the value and the 8x8 board in the format shown above (in red); that is, only the value and the 8 by 8 array of integers. No "attachments", please, and no hit matrices. 

Value = 20.83333...

1 2 3 4 4 3 2 1
2 4 6 8 8 6 4 2
1 2 3 4 4 3 2 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 2 3 4 4 3 2 1
2 4 6 8 8 6 4 2
1 2 3 4 4 3 2 1

Dear Scut,

       What color is the sky in your world?

Your Friend,                                                                                                                               Sandy

the sky is a set of pixels,you could have blue or a range of blues.  or you could have the jimmy hendrix sky.  electric sky, matrix sky. shocking pink,perhaps?

No matter what the color, just, "Kiss the sky"

What's ?

Jimmy knows.-but then, he always did.

And so does Jimi

Solution:

Value = 28.3333

0 1 1 0 1 1 0 0
1 1 0 2 2 0 1 1
0 0 3 0 3 0 2 0
1 3 3 0 3 2 2 1
1 2 2 3 0 3 3 1
0 2 0 3 0 3 0 0
1 1 0 2 2 0 1 1
0 0 1 1 0 1 1 0

Each piece is a distinct type. Its hit matrix is the maximum possible, consistent with its appearance here; that is, it is the most powerful piece that does not attack any of the other occupied squres in this particular setup.

 There are no solutions better than 28.3333 among boards of the following types:

  - All 2^32 boards which are symmetric with respect to reflection
   through the center (aka. half-twist).

 - All 2^32 boards which are mirror self-images.

 - All 2^36 boards which are symmetric about the diagonal.

 - All 31 * 2^36 boards which have 16 1's around the edge.

So any board with higher score would have to be asymmetric.

---

Missed that one didn't I?