This is to outline my spring plan for increasing the number of honey bee colonies. This is mostly derived from Michael Bush's page "A few good queens" You should read this page too. this is just an outline. For all the good info, go read Bush's pages.
Two most important things to get good queens: Cell builder must be densely packed with bees and the comb containing the eggs must be new so the bees can easily tear down the cell wall and feed the larvae well.
Timing: The hive (cell builder) must be strong and ready to draw out new comb in the brood nest. (April sometime - a month after first blooms - ha ha)
Step 0: When hive has enough bees to fill a gap in the broodnest, insert empty frame to get it drawn and eggs laid.
Step 1: Check back in 4-5 days. Goal is to remove the queen as soon as the new comb is drawn has has eggs. The math: An egg takes 3.5 days to hatch. You need to remove the queen from the hive before these larvae are a day old. i.e. 4.5 days from being laid.
Step 2: As soon as you have the eggs/larvae in newly drawn comb, remove the queen along with enough resources and put her in a nuc. NOTE: finding the queen in a densely packed hive is going to be HARD. Use the force. Now is the time to reduce the boxes to minimize the space in the cell builder.
Step 3: 9 days after making them queenless, make up the mating nucs. (Queen math: hatch 3.5, cap: 8, emerge: 16) So if you make them queenless on the 4th day, you want to move them to mating nucs on day 14 from being laid.) If one hatches early she'll destroy all the rest. These nucs really only need a couple hours, so could be done on day 10 if necessary.) Mating nucs need one frame of brood and one of honey.
Step 4: 10 days, (14 from egg) cut queen cells out from comb and place into mating nucs
Step 5: check back in a couple weeks for eggs.