  In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat
K\"ahler metrics on the K3 surface based on a gluing construction. In this
construction, one starts from a flat torus with $16$ orbifold points, and
resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson
manifolds which all have the same orientation. This construction was carried
out rigorously by Topiwala, LeBrun-Singer, and Donaldson.
  In 1981, Page asked whether the above construction can be modified by
reversing the orientations of some of the Eguchi-Hanson manifolds. This is a
subtle question: if successful, this construction would produce Einstein
metrics which are neither K\"ahler nor self-dual.
  In this paper, we focus on a configuration of maximal symmetry involving $8$
small Eguchi-Hanson manifolds of each orientation which are arranged according
to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson
manifolds with opposite orientation, we identify a non-vanishing obstruction to
the gluing problem, thereby destroying any hope of producing a metric of zero
Ricci curvature in this way. Using this obstruction, we are able to understand
the dynamics of such metrics under Ricci flow as long as the Eguchi-Hanson
manifolds remain small. In particular, for the configuration described above,
we obtain an ancient solution to the Ricci flow with the property that the
maximum of the Riemann curvature tensor blows up at a rate of
$(-t)^{\frac{1}{2}}$, while the maximum of the Ricci curvature converges to
$0$.
