Boolean spaces with countable base but uncountably many orbits

We construct two Boolean spaces with countable bases which have continuum many orbits. Two points are in the same orbit iff there is a homeomorphism of the space which carries one to the other. One of our examples is a primitive Boolean space. J. Donald Monk posed these problems. We don't know if there is a countably based Boolean space with exactly ₁ orbits.Presented by R. S. Pierce.