Chess puzzle I found in my dad's old book

Posted by Eswo 2 days ago

Chess puzzle I found in my dad's old book(ardoedo.it)

164 points | 45 comments

dllu 13 hours ago|

Neat. Surprisingly, there are 388 solutions, and a lot of them look rather unintuitive.

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

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

My original intuition was to place the queens on unique rows and columns to cover as much as possible but it turns out there are solutions with three of them on the same row.

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).

sobellian 11 hours ago||

I used CP-SAT to enumerate the solutions. A heuristic for "interesting" solutions is those which only admit one valid bishop placement. For example:

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

Where the bishop lies at the intersection of three queens' horizontal attacks. With these queens, no other bishop placement works.

aaron695 9 hours ago||

[dead]

NooneAtAll3 11 hours ago||

is there a solution where all the pieces are covered as well?

hammock 8 hours ago||

The trick for me was to place a queen (most anywhere, but start with a corner it’s easier), then check and look for the spot with the most reds around it (eg 9, or 8, or 7), place the next queen there, repeat. Then place the bishop as needed.

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.

r3trohack3r 7 hours ago||

I was skeptical but, this not only worked the first time I tried it, this strategy has worked every time.

CamelCaseName 6 hours ago||

Wow!! That was awesome, great explanation too!

tromp 13 hours ago||

> The task is to place four black queens and one black bishop on the chessboard so that there is no square not under their attack. In other words, after arranging the five black pieces, it must be impossible to place the white king anywhere without it being in checkmate.

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.

bryanrasmussen 15 minutes ago||

well there are all sorts of irregularities in the stated puzzle, since you cannot move into check or checkmate it follows that you should not be able to place it on the board and have it be in those conditions.

However let us suppose that you can place it on the board and have it be in either check or checkmate.

If it has been placed on the board and it is in check, since it cannot move into check it follows that it is the king's move. If it can move out of check by taking a piece or another way of moving it is not in checkmate, this seems a pretty weird trick then, because it is not so interesting that you can place the king on the board and it will be in check and it is the king's move to get out of check.

If it can get out of check by taking an unprotected queen then it is again not very interesting, and why would I not just put the king on one of those positions and take the queen from the first.

If I cannot put it on the board without it being in checkmate, it must mean that the placing it on the board is itself the move, and you cannot move into checkmate, but if placing it on the board is the move then it follows you can take a queen with that move if you can place the king anywhere on the board.

The phrasing of the puzzle is inadequate, it seems.

jhncls 11 hours ago|||

Although unlikely, maybe there exists a solution where all black pieces are protected?

yunwal 10 hours ago||

If there is, it's not listed as the solution on the puzzle website

nojs 9 hours ago|||

Yeah, the solution given is actually wrong as stated!

chongli 7 hours ago||

That assumes you can place the king on the board and move immediately. I think the puzzle assumes you can’t do that; that placing your king on the board is your move, thus ending your turn.

billforsternz 6 hours ago|||

The wording is very unambiguous, it means something very specific in chess. In every legal chess position either White is checkmated or Black is checkmated or (by far the most common except in film and TV!) neither side is checkmated. So the wording is crystal clear, you should be able to freely place the White king on any of the unoccupied 59 squares and the position will be one of those in which White is checkmated.

A real shame, this totally ruined the puzzle for me as it seemed so unlikely that all five Black pieces would be mutually protected. I should have forced myself to ignore the faulty clause and try to solve without it. The bad clause is also completely unnecessary - one of those cases where deleting text (or code!) is an improvement with no downside!

Timwi 54 minutes ago||

I'd argue that if you remove the clause you still have the same issue. The problem statement says, “so that there is no square not under their attack”, which in my reading includes the squares occupied by the pieces, and a piece cannot attack itself.

wavemode 5 hours ago||||

There's no legal way to reach a chess position where a king is in check or checkmate and it's not the king's turn to move.

Scarblac 2 hours ago|||

You can only be in check or checkmate when it's your turn though.

tantalor 13 hours ago||

Really nice but I wish I didn't have to click "Check" every time after moving the pieces. It could do that automatically.

hatthew 10 hours ago||

I put `setInterval(checkBoard, 100)` in the console to do it automatically.

xdennis 5 hours ago||

I think that's intentional as in chess you're supposed to come up with the solution without feedback. The puzzle predates computers.

maest 10 hours ago||

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

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.

pimlottc 10 hours ago||

Please don’t use red and green for the colors in the “check” mode, it’s hard to tell apart for colorblind people (especially against the partially shaded black squares).

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.

ElenaDaibunny 20 minutes ago||

this is the kind of HN post i come here for.

hatthew 10 hours ago||

I solved it with this, a pleasingly symmetric solution. I was surprised that a solution exists with all the queens in a row.

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

aidenn0 10 hours ago||

More surprising to me is that there's a solution with no pieces on black.

js8 5 hours ago|||

I started with 4 queens placed symmetrically on c7,g6,f2,b3, which covers everything except corners. Then I shifted all of them diagonally, i.e. to d6,h5,g1,c2. And it turns out, then only a8 and b7 are not covered, which can be easily solved by placing bishop anywhere at diagonal, e.g. h1.

archargelod 10 hours ago||

I found this solution (actually, it's 4 solutions, each B is a different Bishop placement and at the same time it's the only 4 spots on the board not covered by queens):

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

capibara13 49 minutes ago||

I took quite some time with this, but I found it a hard one! Nice job :-)

wrqvrwvq 11 hours ago|

fun demo. could be a daily puzzle combining various commenter suggestions. There are (didn't verify personally) 388 solutions. The daily puzzle could remove 1+ pieces and ask for a 1+ move guess.

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.

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