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Together with your ticket for Inuyama Castle, you will get admission to the Bunraku (Japanese puppet theater) museum just down the road. It makes for a nice walk around seeing how puppets are constructed and operated, you can try moving one puppet by yourself, and puppets are on display.
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犬山城 Inuyamajō
Until it was transferred to Inuyama city in 2004, this was the last remaining privately-owned castle in Japan. It is one of the nicest original examples of feudal Japanese fortifications. It was built in 1537 by Oda Nobuyasu, grandfather of Oda Nobunaga, the warlord who helped end the long civil war that preceded the Tokugawa Shogunate. Inuyama is one of a handful of castles designated a Japanese national treasure. You can have look at the original wooden structure.
有楽苑
Urakuen (Joan Tea Ceremony House) is about 20 minutes walking from Inuyama station, or five minutes walk from Inuyama Castle. It is a national treasure and has a beautiful garden around it. If you want a bowl of green tea while sitting on the veranda of the tea house, add ¥600. If you want to visit Inuyama Castle as well, there is a combination ticket sold here for ¥1300.
明治村
about 20 minutes by bus from Inuyama station, houses a large collection of Meiji-era architecture from all over Japan and even as far away as Brazil, Hawaii, and Seattle. The effect is like that of a ghost town - visitors can walk through the buildings at their leisure, and each one is full of vintage furnishings, as if it were still in use. Fans of Japanese literature may enjoy the summer house of writer Soseki Natsume, where he reportedly wrote his classic novel I Am A Cat. The signature piece, however, is the lobby of the old Frank Lloyd Wright-designed Imperial Hotel (1923-1965) in Tokyo.