Well, I checked the five frame box that I started and found a pretty queen. I guess when I first found two and three eggs per cell, she was just getting started.
Here is my plan, and please correct anything that you see wrong. When I harvest honey this year, I plan to mix in ten empty deep frames, ending up with three deeps on the hive. By doing so, I hope to find the queen moved down into the bottom two deeps. After one week, I plan to move the current five frame into a deep, cover it with news paper, take the top deep off of the three deep hive, and set it on the top of the new queen. Once they cut through the news paper, they should accept her, knowing that they have no eggs of their own to raise a queen from, as they should all be well into larvae stage after one week.
Anything else to consider.
Following this, I plan on feeding both heavily to get them ready for winter. It is funny, I looked the two frames over very well, knowing that they had eggs in them, and capped worker brood, but could not find the queen at all. When I did not find her, I was feeling pretty bad about being a bee keeper, and not being able to find a queen on two frames. I then looked down into the hive only to find her on the bottom board crawling around very nice like.
Funny.