This is easy to do and you will get another colony for very little work.
The crucial element to success with this is to use a good Cell builder hive. There are a wide variety of techniques using starter / finisher hives in a variety of configurations.
I tried several times to use a queenless starter and queenright finisher without success.
As soon as I put the started cells in the queenright finisher the bees would tear them down.
After a lot of research I came across some notes from Michael Bush regarding the use of a queenless starter / finisher and this has worked well for me.
This technique uses only 5 frames, so is quite conservative and efficient use of bees.I like to use a 5 frame nuc box as a cell builder. give them a frame of honey, a frame of pollen, a frame of open brood ( on light fresh comb ) from your selected mother stock, a foundation frame and a frame of sealed brood. Now remove your mother hive from it’s stand and face in the opposite direction
Place your nuc on the mother hive stand.
You will now have a box absolutely packed with bees in a queenless state.
They will build queen cells on the frame of open brood, and get them primed for youif you want to target cell building on grafts or plugs. They may be annoyed and likely to sting .
The reason for using so few frames is to simulate overcrowded conditions that cause bees to swarm, so you will be activating both swarm and emergency impulses for cell building.
After 5 days, you will have sealed queen cells that will emerge around day 16 from the time the day old larvae was started as a cell.
If you are just wanting to produce a single frame of cells to make a single split, now you can just swap the mother hive back to it’s original stand so it has foragers bringing in food again and put the nuc on the new stand, it will pick up a few foragers and the frame of sealed brood will help to give a stable population balance until the queen is mated and starts laying in around 2 weeks. Very small nucs like this may need feeding.
If using a cell plug kit:
On the day after you make up this box, confine your queen mother in the jenter kit in the middle of her brood nest.
On the next day, liberate your queen back into her hive and replace the jenter kit with the eggs facing a frame of open brood.
On the 4th day after the queen is liberated from the jenter kit, remove the frame of brood from your cell builder – it will be covered in capped and nearly capped queen cells,. You can use this frame as is to start a 2 frame nuc, or wait until day 12 and cut out suitable cells to make up nucs from this frame. The sealed queens are fragile at this time so do not shake or invert this frame.
Now you should check your cell builder for any other queen cells on the sealed brood frames and destroy them, so that your target jenter cells are not destroyed by an early hatching queen.
Make up your cell bar with about 15 cells- from the jenter plugs. don’t use more as you are aiming for quality. Insert the queen cell bar where the frame of open brood was.
These larvae will now be the ONLY larvae of suitable age and will be readily accepted.
After 5 days they will be sealed, and you can start another cell bar of cells at this time if wanted.
On day 12 they can come out for distribution to 2 frame nucs. press them into an area with open brood.
If you are just making up frames with queen cells on combs, you can pop a new comb in every weekend and make up a nuc the following week. just check that you do not get any rogue cells being built on the sealed brood you are adding weekly. You can cut cells out of these combs and use in 2 frame nucs as mating hives.
You can keep this cell builder hive going indefinitely by adding a frame or 2 of sealed brood every week to 10 days. inspect for rogue queen cells on the sealed brood frames every 3 days. One virgin queen will destroy your efforts and end the usefulness of this cell builder hive.
If this is all the cells you need, just leave one ripe cell behind in the cell builder and it will become a strong nuc, keep an eye on it for the next few weeks, as the bees may be in the mood to swarm and build queen cells from the first lot of eggs laid by the new queen.
Michael Bush has great information and you will probably find his writing a lot less confusing than mine!
At minimum you will need a veil, hive tool and smoker and a hive, a stand to put them on, and some cans to put the legs in to deter ants.
You could start with a 5 frame nucleus hive, which the bees should soon outgrow or go for an 8 frame colony, which again will rapidly outgrow a single box.
Ideally you should start with two colonies in case something happens to one of your queens, you can add a frame of eggs to the queenless hive and they should make a new queen.