For all we knew, the lifeless little hammerhead sharks were an omen, a fitting enigma for a fabulous place.

My family had just slogged across a coral reef in several feet of water toward the islet known as Chinaman's Hat. Ahead of us, the turquoise of the Pacific stood out against a veil of showers moving in from the sea.

Hammerheads in Hawaii

The hammerheads, babies from the look of them, floated in the gently surging waves near the islet's shore. We had no idea how they had gotten there—a local angler cleaning a larger fish maybe? But I knew for certain that the image—and the mystery—would lock this trip in my kids' memories for all time.

For my part, it would mark my second hike out to Chinaman's Hat; and I have made it a point to get back several times since.

Hawaii offers a full range of opportunities for offbeat adventure, from tame to insane. Through investigation and discovery, savvy visitors can strike out on their own to find special places that are far from the crowds and relatively inexpensive. Such places abound in the backcountry of the outer islands—Kauai, Maui, Hawaii (The Big Island), Molokai, and Lanai. But on Oahu, near Honolulu and the tacky tourist nexus of Waikiki, you really have to follow the lead of the "Kamainas," as the local folks and old-timers call each other, to find choice out-of-the-way fun on the most heavily visited island in the state.

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While Hanauma Bay is very popular, especially among the snorkeling crowd, the Oahu locals favor—and tend to keep to themselves—two small islets: Chinaman's Hat, the aforementioned landmark just up the coast from the windward town of Kaneohe, and Goat Island, an ethereal wildlife preserve off the wild and windy North Shore.

Both islets are endowed with genuine Hawaiian ambiance. They are also distinguished by the fact that they are reachable by foot—over coral reefs—when the tide is right.

Perhaps the best thing about Chinaman's Hat and Goat Island is the mystery; you can never really predict what you might discover on the islets or on the reefs that you cross to get out to them. Each of our visits has been singular in some way—be it the serendipitous delight of a sun shower blowing in from the open Pacific or the eerie appearance of little hammerhead sharks. You just never know.

That hints at another little secret which the Kamainas tend to keep to themselves: Encountering the unexpected is, by far, the best way to really know Hawaii.

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