A puzzle

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.

Edit to add SECOND THOUGHT:
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.

Edited to fix bad arithmetic.

[-]Thomas14y40

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

8, actually.

[-]ArisKatsaris14y00

lol, yeah, thanks.

[-]orthonormal14y30

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.

[-]gRR14y20

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

[-]Thomas14y-10

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.

[-]gRR14y30

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

For two sets, the answer is 118.

[-]Thomas14y10

119.

[-]gRR14y10

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

[-]Thomas14y40

[-]gRR14y00

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.

[-]Thomas14y00

What about the three sets? And four?

[-]gRR14y00

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

[-]TrE14y00

Can you give a position for 53 'bounds'?

[-]gRR14y20

Here

[-]ArisKatsaris14y-10

Then the clarification mentioned in a comment:

"the 16 white pieces" can be mixed colors, doesn't matter.

is false, because by turning the pawn of the second row, and the first pawn of the third row into black pieces, we go up to 55 connections from 53.

[-]Thomas14y00

Makes no sense to me what you said. Can you please clarify?

[-]ArisKatsaris14y10

I mean that the following position:

achieves 55 connections, by using mixed colors. Unless my arithmetic is failing me again.

Edit to add: And unless arithmetic is failing me again, the following has 56 connections:

[-]Thomas14y00

Yes, I know now what you mean. It is a bit different puzzle now. The color of a piece maters this way

Do you think your solution is optimal?

[-][anonymous]14y00

I meant that with mixed colors, we can have the following:

which unless my arithmetic fails me again, is 55 connections.

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[-]RobertLumley14y20

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.

[-][anonymous]14y250

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.

[-]RobertLumley14y20

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

[-]Luke_A_Somers14y00

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

[-]faul_sname14y20

Can the pawns reach the other side of the board?

[-]Thomas14y-10

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.

[-]faul_sname14y00

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

[-]Thomas14y00

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

Sure. It's 196.

[-]jpulgarin14y20

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?

[-]Thomas14y00

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.

[-]faul_sname14y00

Promotion definitely would matter.

[-]Thomas14y00

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

[-]gRR14y10

[-]Thomas14y-10

Very good.

[-]TrE14y00

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

[-]Thomas14y00

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.

[-]TrE14y00

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.

[-]Thomas14y00

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

[-]TrE14y00

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?

[-]jpulgarin14y20

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

[-]Thomas14y00

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

[-]Random83214y00

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

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[-]Thomas14y00

They can be anywhere. The same color, too.

[-]Thomas14y00

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.

[-]BlazeOrangeDeer14y00

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

LESSWRONG
LW

LESSWRONG
LW

-11

-11

-11