CPSC 322 Introduction to Artificial Intelligence

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Presentation transcript:

CPSC 322 Introduction to Artificial Intelligence October 15, 2004

Generic graph search algorithm Given a set of start nodes, a set of goal nodes, and a graph (i.e., the nodes and arcs): make a “list” of the start nodes - let’s call it the “frontier” repeat if no nodes on the frontier then terminate with failure choose one node from the frontier and remove it if the chosen node matches the goal node then terminate with success else put next nodes (neighbors) on frontier end repeat

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2).

CILOG graph search neighbors([2,8,3,1,0,4,7,6,5],[[2,8,3,0,1,4,7,6,5],[2,0,3,1,8,4,7,6,5], [2,8,3,1,4,0,7,6,5]]). neighbors([2,8,3,0,1,4,7,6,5],[[0,8,3,2,1,4,7,6,5],[2,8,3,7,1,4,0,6,5]]). neighbors([2,0,3,1,8,4,7,6,5],[[0,2,3,1,8,4,7,6,5],[2,3,0,1,8,4,7,6,5]]). neighbors([2,8,3,1,4,0,7,6,5],[[2,8,0,1,4,3,7,6,5],[2,8,3,1,4,5,7,6,0]]). neighbors([0,8,3,2,1,4,7,6,5],[[8,0,3,2,1,4,7,6,5]]). : neighbors([2,3,4,1,0,8,7,6,5],[]). neighbors([2,3,4,1,8,5,7,6,0],[]). neighbors([0,2,8,1,4,3,7,6,5],[]). neighbors([2,4,8,1,0,3,7,6,5],[]). neighbors([2,8,3,1,4,5,0,7,6],[]). neighbors([2,8,3,1,0,5,7,4,6],[]). is_goal([1,2,3,8,0,4,7,6,5]).

CILOG graph search search(F0) <- choose(Node,F0,F1) & neighbors(Node,NN) & add_to_frontier(NN,F1,F2) & search(F2). search(F0) <- choose(Node,F0,F1) & is_goal(Node). /* choose(N,Flist0,Flist1) <- append([N],Flist1,Flist0). */ choose(N,[N|Flist],Flist). add_to_frontier(Nodelist,Flist1,Flist2) <- append(Nodelist,Flist1,Flist2).

What if the arcs have different costs? 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

What if the arcs have different costs? 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 2 a 2 1 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 1 b 3 c d 1 3 2 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 2 2 3 3 2 2 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 2 3 1 2 2 3 2 3 2 3 2 1 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

What if the arcs have different costs? 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 2 a 0 2 1 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 1 b 2 3 c 1 d 2 1 3 2 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 2 2 3 3 2 2 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 2 3 1 2 2 3 2 3 2 3 2 1 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

What if the arcs have different costs? 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [a 0] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [a 0] sorted 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [b 2,c 1,d 2] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [c 1,b 2,d 2] sorted 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [b 2,d 2] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [g 3,h 3,b 2,d 2] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [b 2,d 2,g 3,h 3] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [d 2,g 3,h 3] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [e 3,f 5,d 2,g 3,h 3] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [d 2,e 3,g 3,h 3,f 5] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [e 3,g 3,h 3,f 5] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [i 3,j 5,e 3,g 3,h 3,f 5] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [e 3,g 3,h 3,i 3,f 5,j 5] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [g 3,h 3,i 3,f 5,j 5] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [k 5,g 3,h 3,i 3,f 5,j 5] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [g 3,h 3,i 3,f 5,j 5,k 5] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [h 3,i 3,f 5,j 5,k 5] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [m 6,h 3,i 3,f 5,j 5,k 5] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [h 3,i 3,f 5,j 5,k 5,m 6] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [i 3,f 5,j 5,k 5,m 6] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [n 6,i 3,f 5,j 5,k 5,m 6] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [i 3,f 5,j 5,k 5,m 6,n 6] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [f 5,j 5,k 5,m 6,n 6] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [o 5,f 5,j 5,k 5,m 6,n 6] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [f 5,j 5,k 5,o 5,m 6,n 6] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [j 5,k 5,o 5,m 6,n 6] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [l 7,j 5,k 5,o 5,m 6,n 6] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [j 5,k 5,o 5,m 6,n 6,l 7] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [k 5,o 5,m 6,n 6,l 7] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [p 7,k 5,o 5,m 6,n 6,l 7] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [k 5,o 5,m 6,n 6,l 7,p 7] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [o 5,m 6,n 6,l 7,p 7] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [q 7,r 8,o 5,m 6,n 6,l 7,p 7] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [o 5,m 6,n 6,l 7,p 7,q 7,r 8] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [m 6,n 6,l 7,p 7,q 7,r 8] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [y 7,z 8,m 6,n 6,l 7,p 7,q 7,r 8] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [m 6,n 6,l 7,p 7,q 7,y 7,r 8,z 8] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [n 6,l 7,p 7,q 7,y 7,r 8,z 8] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [u 8,v 9,n 6,l 7,p 7,q 7,y 7,r 8,z 8] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [n 6,l 7,p 7,q 7,y 7,r 8,u 8,z 8,v 9] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [l 7,p 7,q 7,y 7,r 8,u 8,z 8,v 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [w 8,x 9,l 7,p 7,q 7,y 7,r 8,u 8,z 8,v 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [l 7,p 7,q 7,y 7,r 8,u 8,w 8,z 8,v 9,x 9] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [p 7,q 7,y 7,r 8,u 8,w 8,z 8,v 9,x 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [s 8,t 9,p 7,q 7,y 7,r 8,u 8,w 8,z 8,v 9,x 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [p 7,q 7,y 7,r 8,s 8,u 8,w 8,z 8,t 9,v 9,x 9] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [q 7,y 7,r 8,s 8,u 8,w 8,z 8,t 9,v 9,x 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [aa 9,bb 8,q 7,y 7,r 8,s 8,u 8,w 8,z 8,t 9,v 9,x 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [q 7,y 7,r 8,s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] * 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [y 7,r 8,s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [y 7,r 8,s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [r 8,s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [r 8,s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [s 8,u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [u 8,w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

frontier: [w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

Review how each one finishes

depth-first frontier: [v,h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

breadth-first frontier: [v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

lowest-cost-first frontier: [w 8,z 8,bb 8,t 9,v 9,x 9,aa 9] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 0 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b 2 c 1 d 2 _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e 3 f 5 g 3 h 3 i 3 j 5 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k 5 l 7 m 6 n 6 o 5 p 7 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q 7 r 8 s 8 t 9 u 8 v 9 w 8 x 9 y 7 z 8 aa 9 bb 8

Comparing blind search strategies All will find a solution in a finite space if a solution exists Depth-first can get trapped in infinite recursion in an infinite space (except in CILOG) Breadth-first and lowest-cost will find a solution even in an infinite space, even if one exists Breadth-first will find path to goal with fewest arcs Lowest-cost-first will find lowest cost path (of course) when arcs have different costs Breadth-first is just depth-first but adding to frontier differently Lowest-cost-first is just breadth-first with more info and sorting

Comparing blind search strategies Breadth-first and lowest-cost-first seem wonderful, but they’re gigantic space hogs in terms of how big frontier can be There’s a whole lot more of this good vs. bad stuff in your textbook (chapter 4.4)...read it Ultimately, all three are expensive approaches to search