Print["Answer:\n", Column@Cases[#, {s_, 0} :> s], "\nNear misses:\n",
   Column@Cases[#, {s_, 1} :> s]] &[{#,
    Count[Boole /@ {Length@# == 12, Total@#[[7 ;;]] == 3,
        Total@#[[2 ;; 12 ;; 2]] == 2, #[[5]] (#[[6]] + #[[7]] - 2) ==
         0, Total@#[[2 ;; 4]] == 0,
        Total@#[[1 ;; 11 ;; 2]] == 4, #[[2]] + #[[3]] ==
         1, #[[7]] (#[[5]] + #[[6]] - 2) == 0,
        Total@#[[;; 6]] == 3, #[[11]] + #[[12]] == 2,
        Total@#[[7 ;; 9]] == 1, Total@#[[;; 11]] == 4} - #,
     Except[0]]} & /@ Tuples[{1, 0}, 12]]