In A* search, when several nodes share the same $f$ value, a tie‑breaking rule determines which state is expanded next. This paper examines nine standard tie‑breaking strategies and, assuming a consistent heuristic, constructs positive‑cost instances that yield an arbitrarily large additive expansion gap between any pair of strategies. A parameterized unit‑cost grid example shows that the expansion‑count ratio between low‑$h$ with FIFO and LIFO can be unbounded. We also study unit‑cost search where non‑goal nodes have $h=0$: exact heuristic values near the goal produce complementary extremal outcomes—low $h$ minimizes the remaining expansions within the perfect region, while high $h$ maximizes the total expansions when every final‑plateau state with $h=1$ is a goal predecessor. Finally, using the weighted evaluation function $f_{\alpha}=g+\alpha h$ and the same $h=0$ condition, any weight $0\le\alpha$ leads to analogous extreme expansion behavior.
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