Two astronomers, Joe and Jane, from different planets in their solar
system meet at an astronomer's convention.  Joe tells Jane that her
planet always appers as gibbous or full phase when viewed from his
planet.  What phases will Joe's planet show when viewed from Jane's
world?  Which planet in their solar system, aside from Joe's home
planet, cannot be Jane's home?

Wish I could answer this one.  Here are some thoughts:

Joe's and Jane's planets are in synchronous orbit around their star. 
Joe's planet orbits inside the orbit of Jane's planet.  Joe's planet
appears to eclipse their star at all times.  From the night side of
Joe's planet, Jane's planet will always appear to be full, but with
the shadow of Joe's planet cast upon it.

Joe's planet will be in a new phase just as the moon is during a solar
eclipse.  Jane's planet cannot be a planet inside the orbit of Joe's
planet.

I'm no astronomer, but to achieve the phases between full and gibbous,
I imagine the answer involves orbiting around a twin star such that
the orbit of Joe's planet is accelerated and decelerated to present
those phases.  This would cause Joe's planet to appear to be crescent
or new.

I wish I'd had homework questions like that!  Have a search for info
on the phases of Jupiter and Venus, and then apply that to Joe and
Jane's observations.

Ian G.

This is an easy one! Jane's planet must always be of higher
orbit(further from the star) than Joe's. Then light from the central
star reflected on Jane's planet as viewed on Joes planet will always
appear in Full phase. (with the rare exception of when the star comes
between Joe's and Jane's planets). If I could draw a diagram, this
would be very obvious.

Your questions.

what phases? All phases, the full spectrum of phases just as the moon
is viewed from Earth.

what planet cannot be Janes home? Well any interior planet to Joe's
else all the phases and not just one will be viewable. Therefore for
example the closest planet to the star cannot be Jane's planet.

Regards,
MSc Astronomy.