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Photo: Pontafon, CC BY-SA 3.0 via Wikimedia Commons
It has a 10.8-m wooden Great Buddha.
Tōchō-ji (東長寺) is a Shingon temple in Hakata, Fukuoka, Japan. Its honorary sangō prefix is Nangakuzan (南岳山). It was founded by Kūkai in 806, making it the oldest Shingon temple on the island of Kyushu.
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博多町屋ふるさと館
gives you a glimpse of the Hakata way of life.
Japan's first Zen temple.
The 8th-century Kushida Shrine is starting point for the annual Gion-Yamakasa Festival.
Hakata-za
An experience worth checking out. Check times and prices at the Hakata-za near Nakasu. If you don't want to stay for the whole show, or don't have so much money to spend, you can watch part of a show for about ¥800; see here for details. Half price student/young (~25) adult tickets may be available day of performance to be released 20 minutes prior to the show; see here for details.
This gallery hosts temporary exhibitions from Kyushu and all around Asia. It has a good shop.