A puzzle

-6 14 April 2012 06:55AM

I have invented it, long ago. It's a test how clever a class or a community is.

Say, you have all white chess pieces on a chessboard. How many connections can they have at the most. A connection is when a piece is "attacking" or "covering" another piece. If this piece is "shooting" back, it's already two connections there between the two.

In the initial position there are 20 connections. 4 by the rooks, 4 by the bishops, 2 by the knights, 5 by the queen, 5 by the king.

Just give me the maximal number.

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Comment author: 14 April 2012 02:13:58PM 2 points [-]

I'm curious as to why people downvoted this so strongly. I wouldn't be surprised to see it at -1 or -2, perhaps, but it seems undeserving of the -6 it had when I upvoted.

Comment author: [deleted] 14 April 2012 02:48:29PM 18 points [-]

I'm sure most of the downvotes are sparked by the line

It's a test how clever a class or a community is.

It turns a potentially-interesting puzzle into a status game. If you don't answer it, it's because you're not clever enough. If you do, you accept that even though you're smart, Thomas is even smarter because he tells you if your answer is good enough.

Comment author: 14 April 2012 04:03:19PM 2 points [-]

Ah, that's a fair point. I didn't really notice that on first read.

Comment author: 19 April 2012 05:44:30PM 0 points [-]

I don't think we'd object if you edited the post to cut it.

Comment author: 14 April 2012 01:48:38PM 2 points [-]

Unless there's some clever structure to the optimal position, this is a problem best solved by computer search. Potentially it could make a good Project Euler problem, potentially not (depending on how much computation the best human-writable program requires to do it).

Also, this post gave the false impression that there was a known answer. That irritated me when I read the comments.

Comment author: 14 April 2012 02:06:02PM 1 point [-]

I think I can prove 53 is the maximum. And the solution does have a nice structure.

Comment author: 14 April 2012 03:28:52PM *  -1 points [-]

Prove it, since your answer is correct, as far as I know.

Can you answer for the case of the two sets of pieces? Black and white, 32 of them. Again the color does not matter.

Comment author: 14 April 2012 05:10:46PM 1 point [-]

No, I'll leave it to you for now.

For two sets, the answer is 118.

Comment author: 14 April 2012 05:14:30PM 1 point [-]

119.

Comment author: 14 April 2012 05:43:35PM 1 point [-]

According to my proof, more than 128-10 is impossible. Are you sure?

Comment author: 14 April 2012 06:01:56PM *  4 points [-]

Comment author: 14 April 2012 07:02:58PM 0 points [-]

Thanks! Clever trick with a rook and two bishops to reduce the length of the top boundary, I missed it.

Cebbs fxrgpu sbe n fvatyr frg: svefg, pbafvqre gjb xavtugf naq n xvat. Rnfl gb frr gurl unir ng yrnfg 4 serr pbaarpgvbaf, ab znggre ubj bgure cvrprf ner cynprq. Gura pbafvqre gur gbc obhaqnel (=cvrprf jvgu ng yrnfg bar serr hcjneq ovfubc-zbir naq ng yrnfg bar serr ebbx-zbir). Gur gbc obhaqnel pna'g or yrff guna 6 cvrprf, rnpu jvgu ng yrnfg bar serr pbaarpgvba, naq ng yrnfg bar bs gur obhaqnel cvrprf unf abguvat va gur ebjf nobir vg, fb vg unf ng yrnfg gjb serr pbaarpgvbaf. Nygbtrgure 64-(4+6+1)=53.

Comment author: 14 April 2012 08:55:09PM 0 points [-]

Can you give a position for 53 'bounds'?

Comment author: 14 April 2012 09:12:49PM 1 point [-]
Comment author: 14 April 2012 09:16:30PM 0 points [-]

What about the three sets? And four?

Comment author: 14 April 2012 10:56:18PM 0 points [-]

Cool! Using the bishops trick, any N-sets for N>2 reduce to N*64 - 8

Comment author: 14 April 2012 10:48:43AM *  2 points [-]

FIRST THOUGHT:

King can hit at most 8 pieces = 8
Queen can hit at most 8 pieces = 8
Each rook can hit at most 4 pieces = 2x4 = 8
Each bishop can hit at most 4 pieces =2x4 = 8
Each knight can hit at most 8 pieces = 2x8 = 16
Each pawn can hit at most 2 pieces = 8x2 = 16

So obviously the true answer can't possibly be above 64 connections.

We can reduce this further. From all columns in which there are pieces, one piece atleast occupies the leftmost column, atleast one piece the rightmost column. These pieces will have their potential attacks limited, atleast by one (if these pieces are pawns), so the true answer can't possibly be above 62 connections.

Comment author: 14 April 2012 11:40:57AM 2 points [-]

Each bishop can hit at most 4 pieces =2x4 = 12

8, actually.

Comment author: 14 April 2012 12:26:13PM *  0 points [-]

lol, yeah, thanks.

Comment author: 14 April 2012 09:17:50AM 1 point [-]

53

Comment author: 14 April 2012 09:26:19AM -1 points [-]

Very good.

Comment author: 14 April 2012 08:01:04AM 1 point [-]

Can the pawns reach the other side of the board?

Comment author: 14 April 2012 08:17:24AM *  -1 points [-]

You can put it anywhere you want it on this 64 squares.

So, that the game could be easily generalized for a "chessboard" of any shape.

Comment author: 14 April 2012 10:09:10PM *  0 points [-]

Can you make them queens, as per the rules of chess?

Comment author: 15 April 2012 06:12:48AM 0 points [-]

Say 7 queens, 11 rooks, 6 knights and a bishop among 20 pawns?

Sure. It's 196.

Comment author: 14 April 2012 07:45:23AM 1 point [-]

So the set of pieces you can use is eight pawns, one queen, one kind, two bishops, two knights, and two rooks?

Can the bishops be placed in squares of the same colour?

Comment author: 14 April 2012 08:14:49AM 0 points [-]

Yes. Put a piece anywhere you want. And there is no "en passant", no "promotion" and no "castling" and no two step pawn move. If anything of these would matter anyhow.

Comment author: 20 April 2012 01:29:22AM 0 points [-]

Promotion definitely would matter.

Comment author: 14 April 2012 07:52:49AM 0 points [-]

Yes. They can be placed anywhere. Pawns on the first or the last line, too.

Comment author: 14 April 2012 08:23:15AM 0 points [-]

Does the problem have a solution that seems obvious in hindsight?

Comment author: 14 April 2012 08:31:09AM 0 points [-]

I don't know. I have a number of contacts possible, but I am not sure is it actually the maximal one. I wonder what numbers will be produced here. 49 for now, is the greatest from this community.

Comment author: 14 April 2012 07:36:28AM *  0 points [-]

Most I got by trying for 3 mins was 49, but that's probably not the max. Theoretical max would be 64, if every piece covered every other.

A technical question: Do the two bishops have to be of different colors?

EDIT: Now 50.

Comment author: 14 April 2012 07:55:21AM 0 points [-]

No. For the game to be easily generalized and 25 knights or whatever be possible to put, there is no color limitation.

Comment author: 14 April 2012 08:14:29AM *  0 points [-]

Wait a minute. We are restricted to the 16 chess pieces available to white, or can we use as many pieces as we like?

Or do you want to hint that one can solve this puzzle by generalization?

Comment author: 14 April 2012 08:19:36AM 1 point [-]

I'm fairly sure we're limited to 16 chess pieces available to white, otherwise the problem can be trivially solved with 64 queens.

Comment author: 14 April 2012 08:25:27AM 0 points [-]

The puzzle is for those 16 initial pieces put anywhere on the chessboard. The color is not an issue.

Comment author: 14 April 2012 08:39:03AM *  0 points [-]

Do the bishops have to be in legal positions? EDIT: already answered

Comment author: 14 April 2012 08:40:11AM 0 points [-]

They can be anywhere. The same color, too.

Comment author: 14 April 2012 08:22:46AM *  0 points [-]

For now just solve the puzzle as it is. The color does not matter, it is there just for the stating explanation. "the 16 white pieces" can be mixed colors, doesn't matter.

Comment author: 14 April 2012 09:41:28AM 0 points [-]

It would matter for pieces like pawns who can only attack and move in one direction, probably.