%?- gtrace, juggle(2, [3]).

%?- juggle(2, [3,3]).

juggle(Hands, []) :- !, Hands > 1.
juggle(Hands, Seq) :-
        Hands > 1,
        all_prods(Seq, 2, Prod),
        Prod1 is Prod + 1,
        try_sequence(0, Prod1, 0, [], Seq, Seq).

try_sequence(N, N, _, _, _, _) :- !.
try_sequence(N0, N, ID0, Air0, Seq0, Seq) :-
        N1 is N0 + 1,
        (   Seq0 = [S|Ss] ->
            (   select(Flying-N0, Air0, Air1) ->
                % re-throw a ball scheduled to arrive at this beat
                (   memberchk(_-N0, Air1) -> fail
                    % two balls at once for this beat
                ;   true
                ),
                Time1 is N0 + S,
                try_sequence(N1, N, ID0, [Flying-Time1|Air1], Ss, Seq)
            ;   % create a new ball
                ID1 is ID0 + 1,
                Time1 is N0 + S,
                try_sequence(N1, N, ID1, [ID0-Time1|Air0], Ss, Seq)
            )
        ;   try_sequence(N0, N, ID0, Air0, Seq, Seq) % repeat period
        ).
        

all_prods([], P, P).
all_prods([A|As], P0, P) :- P1 is A*P0, all_prods(As, P1, P).