The time difference of arrival (TDOA) problem admits exact, purely algebraic solutions for the situation in which there are 4 and 5 sensors and a single source whose position is to be determined in 3 dimensions. The solutions are exact in the sense that there is no least squares operation (i.e., projection) involved in the solution. The solutions involve no linearization or iteration, and are algebraically transparent via vector algebra in Cartesian coordinates. The solution with 5 sensors requires no resolution of sign ambiguities; the solution with 4 sensors requires resolution of one sign ambiguity. Solutions are effected using only TDOA and not, e.g., frequency difference of arrival (FDOA) or angle of arrival (AOA). We first present the 5-sensor solution and then follow with the 4-sensor scenario. Numerical experiments are presented showing the performance of the calculations in the case of no noise, before closing with conclusions. Performance of the calculations is exact within numerical error, and in the small fraction of cases in which source localization does not occur, it is driven by misidentification in resolution of sign ambiguity without priors. We therefore believe the calculations have substantial practical utility for their speed and exactness.