The Goblins Fan Forum

How many bishops will it take to checkmate a king on infinite chessboard?

Are you allowed to set up this infinite chessboard how you like, or is there some starting point where the king is not in check and you move pieces to force a mate?

You move them, like in the real game. Each side has their turn.

In that case you'll need five. You need to cover all the squares the king can move to and put him in check. You could do that with as few as four bishops but doing so requires the king be under check from two different bishops which, since the king can't move into check and you can only move one piece at a time, would be impossible to move into that position. With five bishops it's fairly trivial.

You actually couldn't do it with less than 5. Trying to do it with so little ignores either the fact that the board is infinite or the fact that the king should have the possibility of killing a bishop placed as closely as would be required to attempt to perform the feat with 4 bishops.

That would be the solution with 4, but there's no way to move into this position without cheating. So, yes, 5.

Your turn, Zathyr.

5 bishops and a king walk into a bar... wait, no that's not it.

What have I got in my po- no, no, not that either.

Eh, I'm falling asleep here. New riddle will wait till I wake up. Or if someone else wants to post one feel free.

I think it is 6 but only if they are 3 of each color. I tried tactics with 5, but there was always a sliphole. Except you place the bishops perfectly on a board around the king, which would not happen out of a game. If you place them so they don't threaten the king initially, 5 won't work.

It is more than 5 cause you move in turns with the king and it also depends on the colors. You have to use 2 bishops per line of the trap, since else the king could slip diagonally through it.
So you need 4 to build a large V (like reuse) and then use the remaining 2 for a thrid line to push the king forward into the lowest point of the V until he is check mate. You can also work with two paralell diagonals and move them closer and closer until the king is "squished" between them with no place left to escape.

After I came up with that and read that 5 should be enough, I googled it.
5 is wrong:
Source: http://math.stackexchange.com/questions ... nite-chess

Bishops. Six bishops suffice, with three of each color. White can use a pair of bishops to form a wall, and thus with two pairs white can trap the black king between two walls, which gradually close and deliver checkmate with the fifth or sixth (whichever color is needed). If the black king approaches a bishop, simply move it away on the same diagonal. Five bishops do not suffice, since the black king can simply stay on the color having at most two bishops of that color, and there will always be a square available, since double check will not arise.

Just think about it: You need 3 to trap him inside one of the colors for sure. Since there are two colors, it is 6 for both, not 5, cause he could always run in the color which only has 2 bishops.

D'oh!

Nerre wrote:You can also work with two paralell diagonals and move them closer and closer until the king is "squished" between them with no place left to escape.

Yeah this is what I was thinking but I didn't really think about it. Thought three would be enough to walk the king towards the other line, but that would require bishops constantly switching what color they're on.

Hmm, Nerre has the point. I did not think about that possibility.

Thanks. If somebody else wants to come up with a new riddle, feel free to do it, I got no good one right now.

But nobody stated anything about optimal play. And there are 2 separate directions the king could go in yor example with 4 bishops, Zathyr, NW or SE.

No, it can't. There is a bishop in the upper left corner.

Okay, guess I was looking at it wrong.

Davecom3 wrote:But nobody stated anything about optimal play. And there are 2 separate directions the king could go in yor example with 4 bishops, Zathyr, NW or SE.

No, the problem in his picture is, that the king is set check by TWO bishops. But since you move in turns, this CANNOT happen since this would mean he was in check already the turn before and neither moved out nor got killed by the bishop who set him check. Which just would not happen.

Like I said, I was looking at it wrong. I saw what was going on when I adjusted the spacing of the lines.

Alright something new:

5, 8, 3, 7, 37, ...

What's the next number?

-30?

Nope.

Magical intelligent evil knife?

Nice catch.

You know I was going to work on a MIEK riddle. I'm sure I'll have something eventually, but I figured a little numbermancy will help fill the gap in the meantime.

There is enough information there to determine the next number, but if it helps: the previous number would be -23.

Huh, found it. 100.

Anyone can post next riddle.

It's not 100. I think you may have made a slight arithmetic error, or possibly a typo.