Conformal test martingales are used for testing the assumption of exchangeability, which is central for the validity guarantees of conformal predictors. All algorithms for computing conformal test martingales share the fundamental property of allowing the false alarm rate to be controlled by a user-specified significance level. However, the ability to detect violations of the exchangeability assumption may vary largely across algorithms and distributions. Three algorithms for computing conformal test martingales, i.e., Simple Jumper, Sleeper/Stayer and Sleeper/Drifter, are analyzed and three central components are identified; the investment strategy, the betting function and the wake-up scheme. By combining the options of these components, a total of nine novel algorithms are obtained, complementary to the three previously proposed algorithms. Results from a large-scale investigation using both synthetic and real-world datasets are presented. The results on the former show that while all combinations successfully keep the false alarm rate below the requested significance level, which follows from the theoretical guarantees, the relative effectiveness of the algorithms is shown to depend on the type of distribution shift, i.e., whether there is a change in mean or variance (or both) and whether the change is abrupt or continuous. While the original Sleeper/Stayer algorithm is shown to be the most effective of the twelve considered combinations in a majority of cases, the results also show that the novel combinations are competitive in some cases.