switch(min, max).
switch(max, min).

init(min, 1).
init(max, -1).

next(state(N, P), state(N1, P1)) :-
    N > 0,
    N1 is N - 1,
    switch(P, P1).
    
next(state(N, P), state(N1, P1)) :-
    N > 1,
    N1 is N - 2,
    switch(P, P1).

next(state(N, P), state(N1, P1)) :-
    N > 2,
    N1 is N - 3,
    switch(P, P1).

minmaxvalue(_, [], Acc, Acc) :- !.

minmaxvalue(min, [S | Ss], Acc, V) :-
    value(S, V1),
    V1 < Acc,
    minmaxvalue(min, Ss, V1, V).
minmaxvalue(min, [S | Ss], Acc, V) :-
    value(S, V1),
    V1 >= Acc,
    minmaxvalue(min, Ss, Acc, V).

minmaxvalue(max, [S | Ss], Acc, V) :-
    value(S, V1),
    V1 >= Acc,
    minmaxvalue(max, Ss, V1, V).
minmaxvalue(max, [S | Ss], Acc, V) :-
    value(S, V1),
    V1 < Acc,
    minmaxvalue(max, Ss, Acc, V).

value(state(N, P), V) :-
    findall(X, next(state(N, P), X), Xs),
    init(P, I),
    minmaxvalue(P, Xs, I , V).