How can the following data regarding flower phenotypes be explained?

$$\begin{array}{c|c|c} \hline \text{Parents} & F_1 & F_2\\ \hline \text{Blue}\:\times \text{white} & \text{All blue} & \text{196 blue, 63 white} \\ \text{Blue}\:\times \text{pink} & \text{All blue} & \text{149 blue, 52 pink} \\ \text{Pink}\:\times \text{white} & \text{All blue} & \text{226 blue, 98 white, 77 pink} \\ \hline \hline \end{array}$$

From the above data, it is likely that this is more complex than simple Mendelian inheritance. My guess is that there is epistasis at play here, but I can’t figure out what is really happening. What are the phenotypes of the parents? What genetic loci are involved and which traits are dominant?

(I was thinking blue ($B$) was the dominant epistatic flower color allele—$BB$ or $Bb$ would give blue regardless of the other color allele, and $bb$ would allow the expression of color dictated by the other gene, which has pink ($P$) as its dominant allele. This doesn’t work, as seen in the third row.)

Can anyone lend their aid here?

(The data was taken from here, question 32 in the sample multiple choice.)

Because it seems you might be studying for a test, I will try to not answer directly but show the beginning of a chain of logic that might be more useful.

Let's review the very plain evidence. blue crossed against other colors leads to 3:1 ratios of blue:other. So it seems that blue is dominant in those cases. But when you cross those colors against each other, you get silly nonsense, a 3:1.2:1 ratio or something.

So, based on the linked answers, you are justified in thinking that genetics determines flower color, just because of the simple Mendelian pattern of the first 2 crosses. But things don't work for a straightforward monogenic trait in the pinkxwhite cross.

Let's now analyze your explanation. You say: "I was thinking blue (BB) was the dominant epistatic flower color allele—BB or Bb would give blue regardless of the other color allele, and bb would allow the expression of color dictated by the other gene..."

But look at the parents and offspring of the third cross. You say blue is dominant- but it is in fact expressed in the offspring of non-blue parents! So blue can't both be dominant and be the "controlling" epistatic gene. So now: let's consider some other similar possibilities:

1) blue = dominant and is controlled by another locus (recessive or dominant?)

2) blue = recessive and controls another locus (recessive or dominant?)

Do any of those hypotheses match the data?

A final observation: I find it always helps to add up all the combinations of phenotypes (if possible; for example the ratio of (blue+white) : pink, etc.) to see if anything looks like a familiar ratio.

I hope that is useful (and not too late to be helpful in studying, if that is your goal).

This is an example of recessive epistasis

In short, flower color is controlled by two genes. A recessive mutation can happen in either of those genes. In one gene, the recessive genotype (pp) results in pink flowers. In the other, the recessive genotype (ww) results in white flowers. If neither of genes are homozygous recessive, the resulting phenotype is blue.

When you cross a white flower (ww PP) with a blue flower (most likely WW PP) you get offspring that are homozygous dominant at one locus and heterozygous at the other (Ww PP) and the resulting phenotype is blue. The next generation will have 1/4 white offspring (ww PP) and the rest are blue (1/2 wW PP, 1/4 WW PP). The same pattern is seen for pink flowers.

When you cross a white flower (ww PP) with a pink flower (WW pp), you get offspring that are heterozygous in both genes (Ww Pp) and the resulting phenotype is blue. The second generation shows all possible phenotypes and genotypes.

You do not need to know this much detail to answer that multiple choice question, however. All you need to know is that this is a genetically controlled trait, even if the pattern of inheritance is complicated.