Final answer:
The number of genes affecting a trait with an extreme phenotypic frequency of 1/250 can be estimated as five, based on the pattern that each additional gene doubles the number of genotype combinations and the similarity to the square of a fifth-degree Punnett square.
Step-by-step explanation:
When dealing with inheritance patterns and phenotypic ratios, we can estimate the number of genes affecting a trait through observed phenotypic extremes. For traits governed by additive genes, such as those causing polygenic inheritance, phenotypes often follow a bell-shaped distribution, where the extremes are less frequent. In this case, we observe that the extreme phenotypic categories occur at a proportion of 1/250; this is similar to observing a single phenotype at the 1/16 proportion for a two-gene interaction.
Using the fifth-degree Punnett square (32x32) giving a total of 1024 possible genotype combinations, if one of the extremes is 1/250 of the population, it suggests that there may be at least five genes involved. This is because each additional gene that affects the trait doubles the number of possible genotype combinations (exemplified by a 4x4 grid producing a 16-part ratio).
Therefore, if the extreme phenotype ratio is 1/250, and knowing that a two-gene interaction gives us a ratio of 1/16, then for a one-gene interaction the ratio would be 1/4, following this pattern with each additional gene adding a power of two, we estimate that approximately five genes could be affecting the trait.