Sure, let me explain each statement:

I. The intensity of an earthquake is measured on the Mercalli scale: The Mercalli scale is used to measure the intensity or perceived effects of an earthquake at a specific location. It provides a qualitative assessment of the earthquake`s impact based on observations of people, structures, and the environment.

II. The magnitude of an earthquake is a measure of the energy released: The magnitude of an earthquake is a quantitative measure of the energy released at the earthquake`s source. It is typically determined using seismic instruments and provides an estimate of the earthquake`s size.

III. Earthquake magnitudes are based on direct measurements of the amplitude of seismic waves: Magnitudes are determined by measuring the amplitude of seismic waves recorded by seismographs. Seismic waves are produced by the energy released during an earthquake, and their amplitudes are directly related to the earthquake`s magnitude.

IV. In the Richter scale, each whole number demonstrates a hundredfold increase in the amount of energy released: The Richter scale measures the magnitude of an earthquake. It is logarithmic, meaning that each whole number increase on the scale represents a tenfold increase in amplitude and approximately 31.6 times more energy released. For example, a magnitude 6 earthquake releases about 31.6 times more energy than a magnitude 5 earthquake.

Therefore, the correct statements are I, II, and III.