Scott wonders if a four frame nuc can do well producing its own queen.

Yes, a four frame nuc will do very well, during the swarming season!  Be
certain each nuc has one frame of capped honey and another of pollen.  The
pollen can be either underneath capped honey or fresh, from current year
collection.  One of the other frames should be capped larvae, hopefully just
emerging, and the other should be mostly eggs and 1-3 day-old larvae.

If possible, feed the nuc 1:1 sugar water.  I you can't get an in-hive
feeder into the nuc, consider leaving out the frame of capped honey (sugar
water is better than honey at this stage).  The queen cells will benefit
tremendously if you can feed pollen.  Take a cup of pollen, add enough 1:1
sugar water to make a firm patty.  Smooth the patty down until it is the
thickness of pizza dough.  Place it on a piece of wax paper on top of the
frames and place the cover back on.

If you happen to find a hive getting ready to swarm, wait until the cells
are capped (check daily).  The day they are capped, set up your nucs as
outlined in the first paragraph.  The next day carefully cut out the queen
cells (with a knife).  Do not tip the cells over or lay them on their sides.
With your thumb, make an indentation in a center frame of the nuc, and push
the base of the queen cell into it.  Put the top back on the nuc and close
all entrances for 48 hours.  (That assumes the nucs are not in direct sun,
or have plenty of cross ventilation with the entrances closed.  If either is
not true, leave the entrances open.)

Usually the nuc will readily accept the queen cell, and by using one you get
a laying queen 10 days sooner than otherwise.  No less than 14 days after
you insert the queen cell, check for eggs.  If no eggs appear by day 20, the
bees did not accept the queen cell or the queen did not return from mating.
Expect about 15% of your nucs to not have a laying queen, and plan the
number of nucs accordingly.

Best of luck,

Lloyd
Lloyd Spear, Owner, Ross Rounds, Inc.  The finest in comb honey production.
www.rossrounds.com