[RoughDobermann]

A special kind of mass. A weightless, massless mass. Yeah, that's the ticket.

That explanation has never flown with me. Sounds like an excuse for not thinking, IMO.

[RightWhale]

Photons cannot have mass. Energy, depending on wavelength, but no mass. By Einstein's relativity, if they had mass to begin with, the mass would become infinite when they move at their own chosen natural speed and their gravity would be infinite and they would suck up the whole universe. The universe is not being sucked up by photons, so they must have zero mass. One or the other.