I visited Nokonoshima at the beginning of May, during the "Golden Week" holidays in Japan. Despite being a national holiday, the park was not very busy. The weather was wonderful - clear and sunny and about 18 C, but if you were sheltered from the persistent breeze it felt much warmer. Likewise, when not sheltered or in the shade, the breeze made it feel quite chilly. It's best to pack a windbreaker and fleece top so you can adjust to the changes.
The park admission was only the equivalent of $10 - I think that is pretty reasonable given the beauty of the place and the amount of maintenance required. Besides, its not unlike the cost of admission to a conservation area or national park in North America. Once in the park, it's a must to play a round of Noko-Noko Ball ($5). It's a combination of croquet and golf...and it's a hoot! There are also BBQ facilities where you can bring your own food and drink or you can purchase those from the restaurant.
The island offers a rustic setting separated by just minutes from the hustle-and-bustle of downtown Fukuoka.
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