A puzzle about balancing test tubes in a centrifuge

!!defnk#1#2{leftlangle{#1 atop #2}rightrangle}
defdd#1{nk{12}{#1}}
!!

Suppose a centrifuge has !!n!! slots, arranged in a circle around the
center, and we have !!k!! test tubes we wish to place into the slots.
If the tubes are not arranged symmetrically around the center, the
centrifuge will explode.

(By “arranged symmetrically around the center, I mean that if the
center is at !!(0,0)!!, then the sum of the positions of the tubes
must also be at !!(0,0)!!.)

Let’s consider the example of !!n=12!!. Clearly we can arrange !!2!!,
!!3!!, !!4!!, or !!6!! tubes symmetrically:

Equally clearly
we can’t arrange only !!1!!. Also it’s easy to see we can do !!k!! tubes if
and only if we can also do !!n-k!! tubes, which rules out !!n=12,
k=11!!.

From now on I will write !!nk nk!! to mean the problem of balancing
!!k!! tubes in a centrifuge with !!n!! slots. So !!dd 2, dd 3, dd
4, !! and !!dd 6!! are possible, and !!dd 1!! and !!dd{11}!! are
not. And !!nk nk!! is solvable if and only if !!nk n{n-k}!! is.

It’s perhaps a little surprising that !!dd7!! is possible.
If you just ask this to someone out of nowhere they might
have a happy inspiration: “Oh, I’ll just combine the solutions for
!!dd3!! and !!dd4!!, easy.” But that doesn’t work because two groups
of the form !!3i+j!! and !!4i+j!! always overlap.

For example, if your group of !!4!! is the
slots !!0, 3, 6, 9!! then you can’t also have your group of !!3!! be
!!1, 5, 9!!, because slot !!9!! already has a tube in it.

The
other balanced groups of !!3!! are blocked in the same way. You
cannot solve the puzzle with !!7=3+4!!; you have to do !!7=3+2+2!! as
below left.
The best way to approach this is to do !!dd5!!, as below right.
This is easy,
since the triangle only blocks three of the six symmetric pairs.
Then you replace the holes with tubes and the tubes with holes to
turn !!dd5!! into !!dd{12-5}=dd7!!.

Given !!n!! and !!k!!, how can we decide whether the centrifuge can be
safely packed?

Clearly you