15
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What is the minimum number of princesses you need to place on an 8x8 chessboard so that every empty square is attacked by at least one princess?

A princess is a piece from fairy chess that can move like a knight or a bishop.

(Inspired by recent puzzles)

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21
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Here is the optimal answer with

6 princesses.

Shown as knights;

enter image description here

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  • 2
    $\begingroup$ Ah thank you! This problem was driving me insane. $\endgroup$ – Dmitry Kamenetsky Oct 15 at 5:10
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    $\begingroup$ @Tim If C6 is a queen A5 is still uncovered right ? Its covered if its a knight. $\endgroup$ – GoodSp33d Oct 15 at 9:32
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    $\begingroup$ @GoodSp33d A princess is a piece from fairy chess that can move like a knight or a bishop. $\endgroup$ – pajonk Oct 15 at 12:49
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    $\begingroup$ How easy is it to show that there is no solution with only 5 princesses? $\endgroup$ – Tanner Swett Oct 15 at 14:04
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    $\begingroup$ @GoodSp33d: there are no queens in this puzzle... $\endgroup$ – user10445 Oct 15 at 17:12
9
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I have an arrangement with 9 7? Lot of overlap, but not sure how to improve.

Image of solution:

7 princesses
Dots represent knight moves, lines represent bishop moves.

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  • 1
    $\begingroup$ It's a good start and sets an upper-bound on the solution! $\endgroup$ – Dr Xorile Oct 15 at 0:03
8
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This is a great puzzle that had me really hooked!

Partial answer. I can almost do it with 6:

x are princesses. The only square that remains uncovered is Y. I believe 6 should be possible...

........
........
..x.x...
....x...
.......Y
....x...
..x.x...
........

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    $\begingroup$ You're right about the 6. Thanks for inspiring this! $\endgroup$ – Dr Xorile Oct 15 at 15:38
7
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Brute force (which, mind you, might be faulty) revealed that

no solutions with 5 princesses exist

So the accepted answer is probably optimal. Here are, up to mirrorings and rotations, all solutions I found:

Solution 1

........
.X...X..
...X....
.....X..
........
..X.X...
........
........
Solution 2
........
.X......
...XX.X.
........
........
..X.X...
........
........
Solution 3
........
..X.....
....XX..
........
........
....XX..
..X.....
........
Solution 4
........
........
..XXXX..
........
........
...XX...
........
........

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  • $\begingroup$ My program gives the same results as yours. $\endgroup$ – Jaap Scherphuis Oct 16 at 10:05
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    $\begingroup$ @JaapScherphuis I think we'd all love to see the source code $\endgroup$ – Pureferret Oct 16 at 15:03
4
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In addition to Oray's and Arthur's answers, I also found via brute force that

There is no solution with 5 princesses.

In particular,

There will always be at least three empty squares that are not attacked by any princess in a 5-princess setup, and there are only 4 (or 32, including reflections and rotations) setups for which only three empty squares are not attacked.

The (as close to optimal but still failing) setups:

enter image description here

Code (R):

#######
# Set up chessboard
# reading as book, numbers 1-64
#######
xs <- rep(1:8, times=8)
ys <- rep(1:8, each=8)

get_princess_moves <- function(ind){
  x <- xs[ind]
  y <- ys[ind]

  ### Bishop-like moves (includes self)
  ## Up-right/down-left diagonals
  moves <- c(unlist(mapply(function(xcand, ycand){
    which(xs == xcand & ys == ycand)
  }, x + (-7:7), y + (-7:7))))
  ## Up-left/down-right diagonals
  moves <- c(moves, unlist(mapply(function(xcand, ycand){
    which(xs == xcand & ys == ycand)
  }, x - (-7:7), y + (-7:7))))
  ### Knight-like moves: each combination of +/- 2 in one direction and +/- 1 in the other direction
  moves <- c(moves, unlist(mapply(function(xcand, ycand){
    which(xs == xcand & ys == ycand)
  }, x + c(2,2,-2,-2,1,1,-1,-1), y + c(1,-1,1,-1,2,-2,2,-2))))

  unique(moves)
}

moves_list <- lapply(1:64, get_princess_moves)

n_covered5 <- combn(64, 5, function(inds)length(unique(unlist(moves_list[inds]))))

max_cover5 <- max(n_covered5)

max_inds5 <- which(n_covered5 == max_cover5)

combn(64,5)[,max_inds5]
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  • $\begingroup$ How long does that take to run? Thanks for posting. Great answer $\endgroup$ – Dr Xorile Oct 16 at 19:03
  • $\begingroup$ It takes about 2 minutes on my machine (the vast majority due to calculating the n_covered5 variable). $\endgroup$ – Brent Oct 16 at 19:17
  • 2
    $\begingroup$ @DrXorile (also for Brent, and anyone else interested) Here is some C code that runs in about one second if compiled locally. I can improve readability if you like. $\endgroup$ – Daniel Mathias Oct 16 at 23:55

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