Senkan K.
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This island houses a mountain that is rather high. YES RATHER HIGH. I warned you. Do NOT rent a bicycle; you ABSOLUTELY CANNOT ride it up to the top, you may be able to drag it up and then ride it on your way down, but the scenery is great so you might as well enjoy your walk if you are a tourist(and went through all the trouble to come here AND climb the mountain).
Alternatively you could get a motorcycle if you live in Takamatsu and come here from time to time for a picnic. But for that the island is a bit small.
This is the famous site of the Momotaro legend of Japan and on the top there is the cave of the Oni. Not exactly an attraction but might help you enjoy the island... in summer, especially. It's really cool in the cave(I think it's air conditioned).