  In charmless B -> PPP decays, where P is a pseudoscalar meson, there are six
possibilities for the symmetry of the final state. In this paper, for P=pi,K,
we examine the properties of the fully-symmetric final state. We present
expressions for all 32 B -> PPP decay amplitudes as a function of both SU(3)
reduced matrix elements and diagrams, demonstrating the equivalence of diagrams
and SU(3). We also give 25 relations among the amplitudes in the SU(3) limit,
as well as those that appear when the diagrams E/A/PA are neglected. In the
SU(3) limit, one has the equalities \sqrt{2} A(B+ -> K+ pi+ pi-)_{FS} = A(B+ ->
K+ K+ K-)_{FS} and \sqrt{2} A(B+ -> pi+ K+ K-)_{FS} = A(B+ -> pi+ pi+
pi-)_{FS}, where FS denotes the fully-symmetric final state. These provide good
tests of the standard model that can be carried out now by the LHCb
Collaboration.
