One of the oldest problems in statistics is how to analyze a 2 x 2 table and it is still very much with us today. The case of zero fixed marginals, analyzed via the unconditional non-asymptotic method, is perhaps the one that has received least attention, owing to its computational complexity. This paper presents a critical review of existing publications on the topic, defines the conditions of equivalence, of symmetry, and of convexity to be satisfied by the points in the critical region (and takes advantage of the opportunity to generate, in an abbreviated manner, the sample space), defines two new criteria for ordering the elements in the sample space, analyzes the properties of symmetry of the error c~ (dependent on the two nuisance parameters) 1 (one-tailed test) or to ~ (two-tailed and their impact on the reduction of the parameter space to test) of its size by determining the P-value of an experimental table, shows how to minimize computations by grouping common elements, and finally, it gives various properties which allow the size of the parameter space to be reduced even more, and the number of times by which the size of the test must be determined. For purposes of comparison, it refers to the cases of one or two fixed marginals and to the assertions made by other authors.