Hi Michael,
Your experimental outcome sounds more plausible and is easier to explain than the other one. (but see http://van.physics.illinois.edu/qa/listing.php?id=13496 for why the opposite can also happen)
Dissolving salt in water lowers the temperature at which the water freezes, or at which the ice melts. Salt, when placed on top of a melting ice cube, will dissolve in the little bit of water that melts first, and the dissolved salt lowers the melting temperature of the ice it's in contact with.
Melting ice takes heat energy from the surroundings -- 80 calories per gram of water, salt or no salt. The rate at which the ice melts depends on the rate at which heat energy flows in. Heat flows by conduction, convection, and radiation, and the flow rate depends on lots of external factors. You can change the heat flow rate by adding insulation, by moving the air with a fan, or by changing the temperature of the surroundings, among other things. A properly controlled experiment has all factors the same for the ice cube with the salt and for the ice cube without the salt, so that the effect just of the salt can be determined. If more than one factor is changed, one is never sure which factor caused the difference in the outcome.
Heat travels from hot stuff to cold stuff, and the greater the temperature difference, the faster the heat transfer will be. If the ice has salt on it, it will melt at a lower temperature, and will be colder than ice melting without salt on it. Heat will then travel faster from the warm air to the colder ice than to the warmer ice.
The salt will only affect that part of the ice with which it comes into contact. It may make a little pool of saltwater on the top face of the ice cube, which will "dig into" the ice cube. This is a common observation -- if I sprinkle salt crystals on ice on the sidewalk in the winter, it will look like the salt crystals dig little holes in the ice.
Tom J.
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