Ken's POTW 050531

Safe Chess Pieces

With at least one piece of each type on the board, what is the maximum number of chess pieces you can place on a chessboard, such that no piece is under attack?
If we use the scoring below, what is the maximum score you can achieve?

Pieces and scoring: Queen=7, Rook=6, Bishop=5, Knight(N)=4, King=3, Black Pawn(P)=1, White Pawn(p)=1. Black pawns attack down the board, and white pawns attack up the board. Any piece may occupy any square.

Source: Original. My second choice for the 2005 Puzzle Design Tournament.

Solutions were received from Kirk Bresniker. Are there better ones?

1. With at least one piece of each type on the board, what is the maximum number of chess pieces you can place on a chessboard, such that no piece is under attack?
I do not believe there are any solutions for square board sizes less than 5x5. For boards 5x5 through 8x8, here are the best scores I was able to find with at least one of each pieces used.


nax4.5.all
ALL USED!
|---|---|---|---|---|
| Q |   |   |   |   |
|---|---|---|---|---|
|   |   | w | B | k |
|---|---|---|---|---|
|   | R |   |   |   |
|---|---|---|---|---|
|   |   | b |   | K |
|---|---|---|---|---|
|   |   | B |   |   |
|---|---|---|---|---|
plays=8083 score=32

nax4.6.all
ALL USED!
|---|---|---|---|---|---|
| Q |   |   |   |   |   |
|---|---|---|---|---|---|
|   |   | w | B | k | B |
|---|---|---|---|---|---|
|   | R |   |   |   |   |
|---|---|---|---|---|---|
|   |   | b |   | K |   |
|---|---|---|---|---|---|
|   |   |   |   |   |   |
|---|---|---|---|---|---|
|   |   | k | B | k |   |
|---|---|---|---|---|---|
plays=191140 score=45

nax4.7.all
ALL USED!
|---|---|---|---|---|---|---|
| Q |   |   |   |   |   |   |
|---|---|---|---|---|---|---|
|   |   | K |   | K |   | k |
|---|---|---|---|---|---|---|
|   |   |   |   |   |   | B |
|---|---|---|---|---|---|---|
|   | B | w |   | K |   | k |
|---|---|---|---|---|---|---|
|   | k |   |   |   |   | B |
|---|---|---|---|---|---|---|
|   | B | b |   | w |   | k |
|---|---|---|---|---|---|---|
|   |   |   |   |   | R |   |
|---|---|---|---|---|---|---|
plays=4301767 score=61

nax4.8.all
ALL USED!
|---|---|---|---|---|---|---|---|
| Q |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | w | B | k | B | k | B |
|---|---|---|---|---|---|---|---|
|   | R |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | b |   | K |   | w |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | k |
|---|---|---|---|---|---|---|---|
|   |   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|---|
|   |   |   | k |   | k |   | k |
|---|---|---|---|---|---|---|---|
|   |   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|---|
plays=1205565943 score=82

2. If we use the scoring below, what is the maximum score you can achieve?
I'm assuming here you mean if you don't have to use at least one of each piece, but rather you what the best possible non-attacking scores. Here are the best I've found. Note the similarity of 6x6 ... 8x8.

nax4.2.best
BEST
|---|---|
| k | k |
|---|---|
| k | k |
|---|---|
plays=40 score=16

nax4.3.best
BEST
|---|---|---|
| B | B | B |
|---|---|---|
|   |   |   |
|---|---|---|
|   | B |   |
|---|---|---|
plays=83 score=20

nax4.4.best
BEST
|---|---|---|---|
| k | k | B | k |
|---|---|---|---|
| b |   |   |   |
|---|---|---|---|
|   |   |   | w |
|---|---|---|---|
| k | B | k | k |
|---|---|---|---|
plays=11346 score=36

nax4.5.best
BEST
|---|---|---|---|---|
| k | B | b | B | k |
|---|---|---|---|---|
|   |   |   |   |   |
|---|---|---|---|---|
| K |   | K |   | K |
|---|---|---|---|---|
|   |   |   |   |   |
|---|---|---|---|---|
| k | B | k | B | k |
|---|---|---|---|---|
plays=496781 score=50

nax4.6.best
BEST
|---|---|---|---|---|---|
| k | B | k | B | k | B |
|---|---|---|---|---|---|
|   | b |   | b |   | b |
|---|---|---|---|---|---|
|   |   |   |   |   |   |
|---|---|---|---|---|---|
|   | k |   | k |   | k |
|---|---|---|---|---|---|
| k |   | k |   | k |   |
|---|---|---|---|---|---|
|   | k |   | k |   | k |
|---|---|---|---|---|---|
plays=23215793 score=66

nax4.7.best
BEST
|---|---|---|---|---|---|---|
| B | k | B | k | B | k | B |
|---|---|---|---|---|---|---|
| b |   | b |   | b |   | b |
|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|
| k |   | k |   | k |   | k |
|---|---|---|---|---|---|---|
|   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|
| k |   | k |   | k |   | k |
|---|---|---|---|---|---|---|
|   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|
plays=217747468 score=92

nax4.8.best
BEST
|---|---|---|---|---|---|---|---|
| B | k | B | k | B | k | B | k |
|---|---|---|---|---|---|---|---|
| b |   | b |   | b |   | b |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| k |   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|---|
|   | k |   | k |   | k |   | k |
|---|---|---|---|---|---|---|---|
| k |   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|---|
|   | k |   | k |   | k |   | k |
|---|---|---|---|---|---|---|---|
| k |   | k |   | k |   | k |   |
|---|---|---|---|---|---|---|---|
plays=-1275252740 score=120

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