This time we place white pieces on chessboards of various sizes, that every
empty or occupied cell is attacked exactly four times. It is allowed to use
any number of pieces. Computations get easier compared to triple attack problems
and the resulting solution numbers are small.

This is the only possible pattern with quadruple attacks on a 4*4 board. The
solution is perfectly symmetric.

All eleven possible patterns of chess pieces that attack all cells four times
on a 5*5 board are shown in the next picture. The minimal number of pieces is
nineteen. None of the solutions contains any pawns. It's interesting that a
group of five knights is part of each pattern.

With the 6*6 board there are only eight patterns that attack every cell exactly
four times. The small number is a bit surprising and is even smaller than the
5*5 board number. One of the solutions consists of 26 pieces, all others need
28 to 30 pieces. The minimal solution and three others are symmetric. Six of
the patterns have a nice ring structure of knights in the centre.

Not yet investigated.

There no known extensible solution for quadruple attacks.

All shown minimal 3*n boards are also minimal solutions of the
corresponding n*3 boards. Some of the rectangles have no solutions as has been
proved by exhaustive search.

The following picture shows some minimal solutions for 4*n boards:

Minimal arrangements for diamond shapes:

It is impossible to attack all cells of triangle shapes four times.
It's easy to see, if you try to cover the corner cells.