GRiddles Series C: Puzzle 1

Past forward

The city council is building a museum of the city’s history: an 8 × 8 block of 64 galleries, one for each of the 64 eras the city has lived through, numbered 1 to 64 from the oldest to the most recent. Every side shared by two galleries will be made into either a doorway or a wall. Visitors come in off the street into the entrance gallery, yellow in Figure 1. Six walls have already been built and five galleries have already had their exhibits installed, as shown in Figure 1.

A museum plan is a set of walls containing the six already built, together with a labelling of the galleries by the eras 1 to 64, one to each, extending the five already installed, such that from the entrance there is exactly one route to each gallery that never enters the same gallery twice, and along that route the era numbers increase. The cost of a museum plan is the expected number of doorways a visitor passes walking from the entrance to a gallery chosen uniformly at random among the 64. An optimal wall plan is a set of walls occurring in a museum plan of least cost. How many optimal wall plans are there?

Answer format: Give your answer in base-ten notation.