Quantization, Doppler shift and the invariance of the speed of light via the invariance of the counted numbers of photons: An interesting pedagogical problem

We show that when the observers are located in a plane electromagnetic wave it is not compulsory for them to take into account the time dilation and length contraction effects when the wave is detected from two inertial reference frames in relative motion. We also illustrate the difference between the approach that uses Einstein type of observers confronted with time dilation and length contraction effects and an unconventional approach that uses observers detecting information from the energy that arrives at their location.