  We apply a recently suggested new strategy to solve differential equations
for master integrals for families of Feynman integrals. After a set of master
integrals has been found using the integration-by-parts method, the crucial
point of this strategy is to introduce a new basis where all master integrals
are pure functions of uniform transcendentality. In this paper, we apply this
method to all planar three-loop four-point massless on-shell master integrals.
We explicitly find such a basis, and show that the differential equations are
of the Knizhnik-Zamolodchikov type. We explain how to solve the latter to all
orders in the dimensional regularization parameter epsilon, including all
boundary constants, in a purely algebraic way. The solution is expressed in
terms of harmonic polylogarithms. We explicitly write out the Laurent expansion
in epsilon for all master integrals up to weight six.
