dfs(X, P, [X|P]) :- goal(X).
dfs(X, P, R) :- successor(X,S), notin(S,P), dfs(S, [X|P], R).

goal([0,0,0]).

safe(X,_) :- X =:= 3.
safe(X,_) :- X =:= 0.
safe(X,Y) :- X =:= Y.

successor([X,Y,1], [U,Y,0]) :- X>=2, U is X-2, safe(U,Y).
successor([X,Y,1], [X,V,0]) :- Y>1, V is Y-2, safe(X,V).
successor([X,Y,1], [U,V,0]) :- X>0, X =:= Y, U is X-1, V is Y-1.
successor([X,Y,1], [X,V,0]) :- Y>0, V is Y-1, safe(X,V).

successor([X,Y,0], [U,Y,1]) :- X<2, U is X+2, safe(U,Y).
successor([X,Y,0], [X,V,1]) :- Y<2, V is Y+2, safe(X,V).
successor([X,Y,0], [U,V,1]) :- X<3, X =:= Y, U is X+1, V is Y+1.
successor([X,Y,0], [X,V,1]) :- Y<3, V is Y+1, safe(X,V).

notin(_,[]).
notin(X,[X|_]) :- !, fail.
notin(X,[_|Z]) :- notin(X,Z).

printlist([]).
printlist([X|Y]) :- printlist(Y), write(X), nl.

% the query is 
% ?- dfs([3,3,1], [], X), printlist(X).