The slowing down of the rates of clocks by virtue of their motion (time dilation) and the conclusion that events which are simultaneous for one such clock may occur at different times for another (remote non-simultaneity) are two of the most well-known consequences of Einstein's Special Theory of Relativity (STR). A simple algebraic puzzle is presented which shows that these two predictions are fundamentally incompatible with each other. Both effects are derived from the Lorentz transformation (LT), which is the cornerstone of STR, thereby proving that the theory is not physically viable and is therefore in need of comprehensive revision. Another version of the Lorentz transformation (Global Positioning System-LT) is presented which satisfies both of Einstein's postulates of relativity, but which does away with the space-time mixing characteristic of the original LT. The GPS-LT is in agreement with all experimental data as yet observed regarding the variation of clock rates with motion, as expressed in a Universal Time-dilation Law (UTDL). It is also consistent with the Relativistic Velocity Transformation (RVT) and is therefore able to explain many of the effects previously looked upon as unique successes of the LT.