Illegal cluster
a) Add a black rook and a black pawn
b) Add a black rook and two black pawns
c) Rotate the board 180˚ then add a black rook and two black pawns

a) Add the black rook at c5 and the black pawn at c4. With the pawn removed the last move was a double check.
b) Add the black rook at e3 and the black pawns at d2 and d3. With the d2 pawn removed the last move was Pe4xPd4 en passant, following Pd2-d4. With a black pawn at d2 the e.p. capture is illegal.
c) Add the black rook at f8 and the black pawns at f7 and h6. A similar solution in b) would not work because the last move could be Pc3-c2, therefore the position would not be illegal.

Jacob Hoover: I thought this problem would be difficult to solve, but it was surprisingly easy.

A simpler Dawson illegal cluster problem is white king c3, black king g7 – a) Add two black pawns; b) Add three black pawns.