Hakone is known for its mountain trails, scenic boat and cable car rides, onsen (hot-spring baths) and art museums, including the fantastic Open-Air Museum, an expansive sculpture garden featuring works by Rodin, Miró, and Henry Moore, plus a Picasso Pavilion. It also has killer climbing ropes for kids in a separate playroom called the Castle of Nets. Try to hit the museum early (it's open 9 a.m. to 5 p.m.), to leave time for this next leg: From Gora station, take the $4 funicular to the Sounzan stop (a 10-minute trip), then get on the Hakone Ropeway, a 2.5-mile (4-km) aerial tram that takes you over mountainous terrain and on to Lake Ashi, offering splendid views of nearby Mt. Fuji along the way. At about the midpoint is Owakudani, an area known as the Great Boiling Valley. Stop off there for a hike up the side of a volcano, where you'll see — and smell — the geothermal gases escaping from fissures in the rock. As a snack you can buy eggs that have been boiled in the hot springs below, turning the shells black. It is said that consuming one of these eggs will extend your life by seven years. Either way, they're tasty.

Hakone is a three-hour journey or less from Tokyo's Shinjuku Station on the Odakyu line. To get to the open-air museum from the Odawara or Hakone-Yumoto station in Hakone, take the switchback train to the Chokoku-no-Mori station and it's a few minutes walk from there.

• 1. Hakone Open-Air Museum
  Ninotaira, Hakone-machi, Kanagawa-ken, Tokyo, Japan; 81-(04)-6082-1161 35.2444317139.0503953 hakone-oam.or.jp/eng
• 2. Owakudani Station
  35.244322139.019966
• 3. Lake Ashinoko
  35.209567139.003463