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Here are 3 related chess problems. The color of the pieces is not important. Unlike a legal chess game there are no limits to the number of pieces of each type in these puzzles.

Puzzle 1: On a 4x4 chessboard, place some bishops and some knights (and no other pieces) so that:

Each bishop attacks exactly 3 knights (and nothing else) and
each knight attacks exactly 1 bishop (and nothing else).

Puzzle 2: On a 6x6 chessboard, place some knights and some rooks (and no other pieces) so that:

Each knight attacks exactly 4 rooks (and nothing else) and
each rook attacks exactly 1 knight (and nothing else).

Puzzle 3: On a 9x9 chessboard, place some bishops and some kings (and no other pieces) so that:

Each bishop attacks exactly 3 kings (and nothing else) and
each king attacks exactly 2 bishops (and nothing else).

Attribution: Erich Friedman

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Solutions to all 3:

enter image description here

You can actually use the requirements to create ratios that help solve the puzzle:

For instance, in the first, each bishop attacks exactly 3 knights, whereas each knight attacks 1 bishop, so the ratio of knights to bishops will be 3:1. This means the solution will be a multiple of this.

Furthermore for the knight puzzles; due to the way the knights attack, if it is attacking a bishop or a rook it won't be under attack from that same rook or bishop. This means in both puzzles 1 and 2, there has to be at least 2x the ratio to solve - and in both cases the solution is exactly 2x.

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  • $\begingroup$ @Lucenaposition fixed $\endgroup$
    – Beastly Gerbil
    Commented Aug 26 at 23:34
  • $\begingroup$ +1 Your doubling argument about the knights is clever. $\endgroup$
    – Will.Octagon.Gibson
    Commented Aug 27 at 0:01
  • $\begingroup$ Your logic does not apply in general. On larger grids, we can find solutions with different ratios. $\endgroup$
    – Daniel Mathias
    Commented Aug 27 at 4:41
  • $\begingroup$ Indeed (re:@DanielMathias), it appears you can remove one or two of the "inner" kings from the solution to puzzle 3 and break the ratio? And in puzzle 1, on a larger board, you could add additional knights attacking the bishops while not being attacked themselves? $\endgroup$
    – Ben Reiniger
    Commented Aug 27 at 13:37

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