If you are into culture, do this bounce in 2.5 weeks:
1) Beijing - 3 days 2 nights (must sees - Great Wall, Forbidden City)
2) Tokyo - 3 days 2 nights (must sees - Tokyo Imperial Palace, National Museum)
3) Taipei - 3 days 2 nights (must sees - National Palace Museum, Chiang Kai Shek Memorial Hall, Sun Yat Sen memorial Hall, Taipei 101)
4) Hong Kong - 3 days 2 nights (must sees - Victoria Harbor, The mountain PEAK, the tram)
5) Chengdu - 3 days 2 nights (must sees - Panda base, Zhuge Liang house, traditional Szechuan change-face show)
That's 15 days, I would say those are good jumps to get a taste of Chinese/Japanese culture