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OK so the latent heat of the cold ice cubes goes with time, this is because of convection, now with time the cold ice cube when kept in a warm temp(the temp above 0 degree Celsius)the coolness of that cube starts to flow,now if that same ice cube is put in water the ice cube will start to lose its coolness and the water will start to lose its warmness and this keeps on happening until the temp of water and the temp of water is equal 
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Simply put as ice cubes have temperature way lower than water transfer of heat takes place through conduction.the temperature becomes stable at around 10 to 20 degree Celsius which is lower than room temperature. Hence we feel cool when we touch it.
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