Chess puzzle I found in my dad's old book (opens in new tab)

    ........
    ...Q....
    ........
    ........
    .....Q..
    ........
    ........
    Q..B..Q.

    Q.......
    ........
    ........
    ........
    ..QQB..Q
    ........
    ........
    ........

Python script: https://gist.github.com/dllu/698d5f71b2b9735c5c462ddf4a2f6fc...

Here's how it works:

0. precompute the attack patterns of each possible queen/bishop location as a bitmask, stored as an integer

1. generate candidate solutions, allowing attack rays to pass through other pieces, by brute forcing the positions of the 5 pieces and taking the bitwise OR of their attacks

2. out of the candidate solutions, check which ones are actually valid taking into account occlusion. Actually, you only need to check if the queen's horizontal attack is blocked by the bishop, as queens cannot block each other (the blocking queen herself has the same attacks so they effectively pass through each other).

That last word should be "check". not "checkmate". A king next to an unprotected queen will be in check but not checkmate as it can capture the queen.

The key was realizing the proximal spaces next to the placed queen are the most important to cover. Forget about trying to have a long reach, it comes naturally.

Bonus problem: find an arrangement of 4 queens on the board such that:

1. There is exactly 1 square on which a bishop can be placed, such that the 4 queens and the bishop attack all unoccupied squares

AND

2. There is exactly 1 square on which a rook can be placed, such that the 4 queens and the rook attack all unoccupied squares

Amazingly, modulo rotation / reflection, this problem has exactly 1 solution.

I think this is a bit ambiguous and, strictly speaking, wrong for the solution as given.

In particular, this asks for the king to be in check _mate_. Does this require all black pieces to defend each other? Otherwise, white king on the board would not be in checkmate if you place it next to a queen and can immediately capture.

From the solution, you can see it's not a checkmate requirement, just a check requirement.

In fact, there isn’t really a need for two colors. Just color the squares that are threatened by the pieces and leave the rest blank. The meaning will be obvious.

I remember being given another puzzle in my childhood, where the task was to put 8 queens on the board and none of them should "see" each other. Wonder how many solutions exist for that one..

ETA: Apparently it's a classic puzzle, and there are 92 solutions - https://en.wikipedia.org/wiki/Eight_queens_puzzle

    . . . . . . . .
    . . . . . . . Q
    . . B . . . . .
    . . . . . Q . .
    . . . . . . . .
    . . . Q . . . .
    . . . . . . . .
    . Q . . . . . .

I didn’t find it that difficult. A good (?) guess is that the queens would be in (rotational) symmetry, and hence form a square. To maximize coverage, the angle of course is neither 90° nor 45°, and the queens will have a minimum distance of 3 from each other. Going from smallest distance possible between queens to largest, one tries (3, 1), (3, 2), (4, 1), done. One might not notice that (4, 1) works (which happened to me), because it only works when the square is at the edge of the playing field. But the next one, (4, 2), works unconditionally. Or going from largest to smallest, one tries (6, 1), (5, 2), (5, 1), (4, 3), (4, 2), done.

- ChatGPT was able to solve it in 6 seconds.

- Opus exhausted my entire days token limit on the problem & didn’t solve it.

Also a click on a square could auto place a queen and a second click would swap to the bishop. Every click could auto-check.

A separate discovery mode could start blocking out the squares visually as you place pieces. For a lot of people, that would be easier than the mental representation.

    ...Q....
    .......Q
    ........
    ........
    ..Q.....
    ......Q.
    ........
    B.......

> In other words, after arranging the five black pieces, it must be impossible to place the white king anywhere without it being in checkmate.

These two sentences mean very different things in the normal rules of chess. And if you replace the word “checkmate” with the word “check” in the second sentence it still doesn’t mean the same thing as the first sentence.

The first sentence implies that all the pieces must be defended.

Edit: Eh, I guess it depends on how you view the word “attack” since all the pieces are the same colour.

https://knight-queen-game.netlify.app/

Should maybe update the instructions to clarify that the dark-squared bishop is not constrained to dark squares.

My friend code on BadukPop is EGVNY if anyone wants to play together!