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I like the puzzle How many Chess Pieces are needed to control every square on the board? No Piece Restriction asked 11 years ago.

Here is a variant

Control every square on the board and use as few duplicates as possible.

You may use as many pieces as you like. Pieces do not control the square they occupy.

The king may be used as many times as necessary.

Accepted Answer goes to the person that has the least score.

The score of the following solution is: 3 duplicate pawns + 7 duplicate bishops = 10.

enter image description here

If you use a piece only once, it is not duplicated and the associated score is 0.

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  • $\begingroup$ Please check the tag description for [logic-grid]. It is used for a very specific type of grid puzzle also known as an Einstein's Riddle, not for any puzzle involving logic that takes place in a grid. $\endgroup$ Commented yesterday

2 Answers 2

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![enter image description here

Here is my answer as a score of

2

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    $\begingroup$ I don't think a1 is covered (note that the rook does not control its own square) $\endgroup$ Commented yesterday
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    $\begingroup$ can easily add King to b2 to resolve that without changing the score. But yeah. $\endgroup$ Commented yesterday
  • $\begingroup$ yes, I will fix that, just looking for an answer as 1 if it is possible at the moment, but I dont think so :) thanks for the feedback $\endgroup$ Commented yesterday
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    $\begingroup$ I confirm optimality. Here's another one: $$\begin{matrix}&R&.&.&.&.&.&.&.\\&.&Q&.&.&.&.&.&.\\&.&Q&.&.&.&.&.&.\\&.&.&.&.&.&.&.&.\\&.&.&.&P&.&B &.&.\\&.&.&.&.&N&.&.&.\\&.&.&.&K&.&.&.&Q\\&.&.&.&.&.&.&.&.\\\end{matrix}$$ $\endgroup$ Commented yesterday
  • $\begingroup$ @RobPratt what's the P(awn) for? Seems like any square attacked by it is already attacked by other pieces, regardless of the orientation. $\endgroup$ Commented 22 hours ago
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I just opened the book "Chess kaleidoscope" by Anatoliy Karpov and Evgeniy Gik, and the first diagram which I saw

was that by Sam Loyd, where three queens and two rooks control the chessboard

Three queens and two rooks control the chessboard by Sam Loyd

So it has the score 3.

On the other hand, we can show that the least score is bigger than zero as follows.

Suppose for a contradiction that a set of distinct pieces controls every square of the board. We assume that pieces are transparent, that is if a piece stays at the control line of a long-ranged piece, then it blocks from the control no squares of the line. Then we place the missing pieces at any free squares of the board, keeping the board control.

Suppose first that the queen and the bishop are placed at the squares of distinct colors.

Let C be the color of the knight square. Then there are 32 squares of the color C, but it can be shown that among them are controlled at most 25 by the queen, the bishop, and the rook, at most 4 by the king, and at most 2 by the pawn, so at most 31 squares in total, a contradiction.

Suppose now that the queen and the bishop are placed on the squares of the same color. Let C be the other color.

Then there are 32 squares of the color C, but among them are controlled at most 8 by the queen, at most 8 by the rook, at most 8 by the knight, at most 4 by the king, and at most 2 by the pawn, so at most 30 squares in total, a contradiction.

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