Understanding Genetic Dynamics of Specific Problem

The following questions comes from International Biology Olympiad 2014, theoretical part B.

It states:

Earlier research suggested that the red flower pigment of plant species was the result of a chemical pathway including multiple steps and that all intermediate pigments were white. Three pure-bred lines with white flowers (White 1, 2 and 3) of this species were crossed with each other and the following ratio of colors were obtained among the progeny: The question was how many genes control flower color. I completely botched it. Here are my thoughts:

Looking at F2 generation, there are 9 reds + 7 whites = 16 total. (I know it's a ratio.) Generally, having 16 suggests at 2 genes controlling the phenotypes. For example, cross AaBb x AaBb, you'll get 16 phenotypes.

Turns out 2 is incorrect, and I have no clue as to how colors are inherited in this problem. Any suggestions?

First of all, since two is incorrect, and one is not possible, three is the most probable answer. It is also supported by the format of question that three different types of white purelines are given. Let the genes be A, B, C. Now, there are three possible ways:-

1) Each one has two homozygous dominant genes and one homozygous recessive.

2) Each has one homo dominant and two homo recessive.

3) Any one has all homozygous recessive and the rest two can have any of the above two.

The problem with (2) & (3) is that they do not explain the red F1. Red will require a dominant allele of each gene. But in crosses of (2) or (3), at least one gene will be left out. For example, AAbbcc x aaBBcc will give all progeny with cc. So none can be red. Same problem with aabbcc x aaBBcc or AABBcc

So lets go with (1). Let

W1 AABBcc

W2 AAbbCC

W3 aaBBCC

W1 x W2. => all AA Bb Cc (Red F1 of cross 1)

F1 x F1 will be like a cross of BbCc with AA in front of all the genotypes. This will give you exactly 9:7 phenotypic ratio as mentioned in question.

Now, lets solve the question.

A) Three genes. So false

B) F1 were homozygous for one gene and heterozygous for other two. So false

C) F1 of cross 1 = AABbCc and W3 aaBBCC Since it is evident that it is possible to inherit dominant alleles of each gene (eg. AaBbCc or AaBbCC) so red colour can come. So false

D) F1 cross 1 = AABbCc

F1 cross 3 = AaBBCc

It is evident that one segregation of gene alleles of “a” and “b” will have a dominant allele no matter what. So for a white colour, only chances are from cc. So in cross Cc x Cc there is 1/4 probability for cc. So all in all, 1/4 probability for white (since aa and bb genotypes are not possible here).

So D is True.