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There is a third strategy and that is to study what heuristics a human player uses.


You forgot to calculate and take into account the costs of the actual paths. You forgot to accumulate the cost of the edges for going forward and backward multiple times! The evaluation function of uniform-cost search (UCS) is $f(n) = g(n)$, where $g(n)$ represents the cost of the path from the start node to $n$. The evaluation function of A* is $f(n) = g(n)...


As you found $N$ is the number of nodes that are expanded. The cost of expansion of each node is equal to the number of children of that node. Hence, we use $b^*$ for each node. In other words, the total number of nodes that are involved in the expansion process is $N \times b^*$.

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