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Photo: Pontafon, CC BY-SA 3.0 via Wikimedia Commons
Japan's first Zen temple.
Shōfuku-ji (聖福寺) is a Buddhist temple located in the Gokushōmachi neighborhood of Hakata, Fukuoka, Japan. It belongs to the Rinzai school Myōshin-ji-branch of Japanese Zen. Its sangō prefix is Ankokuzan (安国山). It was founded by Eisai with support from Minamoto no Yoritomo, and is the oldest Zen temple in Japan. Its precincts were designated a National Historic Site in 1969.
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博多町屋ふるさと館
gives you a glimpse of the Hakata way of life.
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.