An Illegal Cluster is an illegal position that becomes legal
as soon as one piece (except a King) is removed.
T. R. Dawson (1889-1951) invented the concept in 1933. By extension,
the term denotes a genre of construction problems, where the stipulation
asks to build an Illegal Cluster, most often by adding a given set of
pieces to a given position.
Variants of the basic notion are White Illegal Clusters (resp.
Black), which become legal as soon as one White (resp. Black) piece
is removed. Similarly, Legal Clusters are legal positions which
become illegal when any piece (even a King) is removed.
Here is a simple example:
T. R. Dawson
The Problemist Fairy Suppl., 1933
1+2. Add 1 bl. Bishop and 1 bl. Rook to obtain an
Solution: add black Bh8 and Rg4. The wK is under
an impossible triple check, but as soon as one piece is removed the
remaining double check is explainable.
An Illegal Cluster (Rex Multiplex) is an illegal position which becomes
legal after one piece (except the last king of one color) is removed.
See also Rex Multiplex.