Observations and results
Did the egg float in cup 1 and 2, but not in cups 3, 4 or 5?
You likely saw that the egg floated best in cup 1, floated a little less in cup 2 (but part of it was above the surface) and did not float in the other cups. Cup 1 had the undiluted salty solution that you originally prepared, which was one half cup of salt in two and one half cups water total. The concentrations of the salt solutions in cups 2 to 4 were halved as you increased in cup number; for example, the concentration of the salt in cup 2 was half that of cup 1, and the concentration of the salt in cup 3 was half again of cup 2. (Cup 5 had plain tap water.) The egg should have sunk in cups 3, 4 and 5 because the density of the egg was higher than the density of the solutions (or plain tap water) in those cups. Cups 1 and 2 had more salt in them than the other cups (with cup 1 having the most salt), which means these solutions were denser. The egg should have floated (with part of it above the water surface) in these two cups because the solutions were denser than the egg. The actual density of the egg is in between the density of the solution in cup 3 and that in cup 2.
More to explore
What Is Density?, from Charles E. Ophardt, Elmhurst College
Why Is the Ocean Salty?, from Herbert Swenson, U.S. Geological Survey Publication
Fun, Science Activities for You and Your Family, from Science Buddies
How Salty Does the Sea Have to Be for an Egg to Float?, from Science Buddies