  Traditional excursion set based models of H II bubble growth during the epoch
of reionization are known to violate photon number conservation, in the sense
that the mass fraction in ionized bubbles in these models does not equal the
ratio of the number of ionizing photons produced by sources and the number of
hydrogen atoms in the intergalactic medium. E.g., for a Planck13 cosmology with
electron scattering optical depth $\tau\simeq0.066$, the discrepancy is
$\sim15$ per cent for $x_{\rm HII}=0.1$ and $\sim5$ per cent for $x_{\rm
HII}=0.5$. We demonstrate that this problem arises from a fundamental
conceptual shortcoming of the excursion set approach (already recognised in the
literature on this formalism) which only tracks average mass fractions instead
of the exact, stochastic source counts. With this insight, we build an
approximately photon number conserving Monte Carlo model of bubble growth based
on partitioning regions of dark matter into halos. Our model, which is formally
valid for white noise initial conditions (ICs), shows dramatic improvements in
photon number conservation, as well as substantial differences in the bubble
size distribution, as compared to traditional models. We explore the trends
obtained on applying our algorithm to more realistic ICs, finding that these
improvements are robust to changes in the ICs. Since currently popular
semi-numerical schemes of bubble growth also violate photon number
conservation, we argue that it will be worthwhile to pursue new, explicitly
photon number conserving approaches. Along the way, we clarify some
misconceptions regarding this problem that have appeared in the literature.
