Hinted by "evil / influence", these puzzles are not actually chess, but Masyu. But they are "impossible positions" as stated, in fact each has a relatively obvious obstruction. For example, the ninth one has a white in the lower corner, which cannot be done in Masyu. On each board, a single chess move (by white) must be performed to make a (uniquely) solvable Masyu puzzle.
Board 1
Obstruction: two blacks in the upper corner. To fix, white needs to capture one of them – and obviously not the one in the corner, since then there would be a white in a corner. The rook can be captured with the knight or the bishop, and it's the latter that gives us a solvable Masyu. Bxh7.
Chessboard
Masyu
Board 2
Obstruction: whites along the bottom edge interfere with each other. One of them needs to move. No move of these pieces within the first rank leads to a solvable Masyu. Moving the king also doesn't, neither does moving the rook up one space. Moving the rook two spaces does work: Re3.
Chessboard
Masyu
Board 3
Obstruction: the white on the top edge forces the black to extend down and left … forcing the same of the next two blacks … which then effectively traps the next white into a corner. Capturing the first of these blacks would resolve the obvious problem, as would moving the pawn. Of these, the only one that actually gives a solvable Masyu is to advance the pawn, to c6.
Chessboard
Masyu
Board 4
Obstruction: white has 3 (or more) in a row in two directions. To fix, we would need to move either the pawn in front of the king (but it can't move), the pawn next to the queen, or the bishop. Of these two remaining pieces and their available moves, it's not obvious, but the only move that gives a solvable Masyu is Bf5.
Chessboard
Masyu
Board 5
Obstruction: two adjacent blacks run into a white along the edge. To fix, one of the blacks must be captured or the white must move. Capturing either black pawn gives no Masyu solutions. Moving the rook along the rook file gives no solutions. The only position with a solution comes from moving the rook one space to the side, Rb5.
Chessboard
Masyu
Board 6
Obstruction: three whites along the edge. To fix, one of them must move. The pawn moves also lead to unsolvable Masyu. The king could take the knight, but that doesn't give a solvable Masyu. Moving the king next to the rook would, but that would put the king in check and so is not a legal move. Of the rook's many possible moves, the only one giving a solvable Masyu is Rd4.
Chessboard
Masyu
Board 7
Obstruction: three adjacent blacks. To fix, one of them must be captured. This one is more difficult to explore, but the only position with a Masyu solution comes from Bxc3.
Chessboard
Masyu
Board 8
Obstruction: black on the edge runs into a pair of whites. To fix, one of those whites must move or the black must be captured. The black piece is not under attack. The pawn's only move does not give a solvable Masyu. Two squares around the king give a uniquely solvable Masyu, but one is an illegal chess move (f4, into check), and so the answer is Ke6.
Chessboard
Masyu
Board 9
Obstruction: white in the lower corner. The obvious thing to try is to move that rook, but in fact trying to solve any of the resulting grids traps the queen in a loop by herself. The correct move is to castle! O-O. Since castling is technically a move of the king (most usually relevant to touch-move rulings), this will be counted later as a move to g1.
Chessboard
Masyu
Extraction
Then, for each board, take the square that was moved FROM, and the square that was moved TO, and extract the corresponding positions from the 8x8 arrangement of letters at the bottom of the page. Read in puzzle order, the FROM and TO sequences will spell out the messages NEXT PIECE/CLOCKWISE. This is an instruction to move along each solved Masyu loop (so that solving the Masyu are actually needed for the solve), from the moved piece, clockwise around the shape enclosed by the Masyu loop, and finding the next chess piece (i.e. Masyu-clue circle) along the loop. Extracting the letters at those locations, again read in puzzle order, we read MEGASTUDY. (A chess problem like these is often called a "study", and the flavortext said you could do a million of them.)
| Board # | Chess Move | Move Extraction | Next Piece | Piece Extraction |
|---|---|---|---|---|
| Board 1 | Bd3-h7 | N-C | Nh5 | M |
| Board 2 | Re1-e3 | E-L | Pf2 | E |
| Board 3 | Pc5-c6 | X-O | Pd6 | G |
| Board 4 | Be4-f5 | T-C | Rf8 | A |
| Board 5 | Ra5-b5 | P-K | Ba6 | S |
| Board 6 | Rh4-d4 | I-W | Kd5 | T |
| Board 7 | Be1-c3 | E-I | Pf3 | U |
| Board 8 | Kf5-e6 | C-S | Pf6 | D |
| Board 9 | Ke1-g1 | E-E | Rf1 | Y |