Thermocol Box is a bad conductor of heat so it does not allow the entry of heat from the walls of thermocol box from surrounding to the box in which ice cubes are kept. Therefore the Ice Cubes stored in a thermocol box remain in solid state for a long time. Thermocol box acts as an insulator.
Ice stays Ice. Because of the only reason that in thermacol box there isn't any temperature that could melt ice.