Question

ice melt but gauge not melt why

Answer 1

you mean that it is possible that under certain circumstances that the melting of the ice could not yet be complete, when the pot as a whole has reached a rolling boil. This is certainly possible, but its feasibility will depend on the strength of the heat applied, the size (and temperature) of the chunk of ice, the density and number of layers of the wire gauze and the volume and initial temperature of the water.

Of course, if I am allowed to choose those variables at will, I can achieve the same thing without the gauze (start with almost boiling water, apply a very high heat with a large chunk of ice in a large pot of water.

Basically, the boiling of the water is a balance between the heat input from the stove and the heat absorption by the ice at it melts (due to the latent heat of fusion). If the input exceeds the heat extraction, the temperature will rise. If it rises fast enough, the water can reach boiling before the ice is totally consumed.

The wire gauze will greatly reduce the rate of melting of the ice by suppressing convection, which in a boiling pot is by far the most important mode of heat transport between the bulk of the water and the piece of ice.