Just after making the case for waves above, I found the following paper at:
Use of Hamilton's canonical equations to rectify Newton's corpuscular theory of light: A missed opportunity
Authors: Robert J. Buenker
Comments: 16 pages, 1 figure
Subj-class: History of Physics; General Physics
Journal-ref: Khim. Fys. 22, No. 10, 124 (2003)
The erroneous prediction of the speed of light in dispersive media has been looked upon historically as unequivocal proof that Newton's corpuscular theory is incorrect. Examination of his arguments shows that they were only directly applicable to the momentum of photons, however, leaving open the possibility that the cause of his mistake was the unavailability of a suitable mechanical theory to enable a correct light speed prediction, rather than his use of a particle model. It is shown that Hamilton's canonical equations of motion remove Newton's error quantitatively, and also lead to the most basic formulas of quantum mechanics without reference to any of the pioneering experiments of the late nineteenth century. An alternative formulation of the wave-particle duality principle is then suggested which allows the phenomena of interference and diffraction to be understood in terms of statistical distributions of large populations of photons or other particles.
Full-text: PDF only
Now it appears that "large populations of photons or other particles" are a requirement.
So it would seem that the one-at-a-time particle would still fail the double slit case.
However, this brings to mind what Feymann did. He claimed that the double slit experiment could be understood completely in terms of particles only if we allowed for anti-particles to return from the future. That was how he developed QED.
So I quess we have a choice. We can either believe that particles exist with a flux of forward time moving interacting with an equal and opposite flux of backward time moving anti-particles; or we can believe that particles do not exist. I prefer to believe that particles do not exist, and that the future is unknown.