Best activities: 1-15
Also known as Akasaka Palace, this neo-Baroque beauty dates back to 1909 and is where the Japanese government receives foreign dignitaries. It's recently been opened to the public on a limited basis: the Front Garden can be entered for free on official opening days, while advance applications and an entrance fee are required for the Main Building and Garden. You'll need to apply at least a month in advance, so plan ahead – and see the full details on the official website.
Best activities: 16-30
Finding tickets to see the big guys in action at Ryogoku Kokugikan can be a bit of a wrestle. Instead, you can visit a sumo stable to watch an early morning practice for free. The catch is that you have to wake up at the crack of dawn, as practice usually starts around 6.30am in the stable.