In a species of rodent, fur colour may be black or brown and their tail length either long or short.

A pair of rodents were crossed, and their progeny then allowed to interbreed. The result in the F2 were as follows:
29 with brown fur with long tails; 9 long tailed with black fur; 10 short tailed with brown fur; 3 short tailed with black fur;
Using the information provided, explain these results as fully as possible justifying your answers with written text and genetic diagrams (with Keys) to show your logic as you proceed backwards through the generations of rodents.

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To understand the results of the F2 generation and determine the possible genotypes of the parents, we need to analyze the patterns of inheritance for fur color and tail length.

Let's consider the genes responsible for fur color and tail length:
1. Fur color: We'll use the letter "B" to represent the dominant allele for black fur and "b" for the recessive allele for brown fur.
2. Tail length: We'll use the letter "L" to represent the dominant allele for long tails and "l" for the recessive allele for short tails.

Based on the F2 results given, we have the following observations:
- 29 rodents had brown fur with long tails (bbLL).
- 9 rodents had black fur with long tails (B_L_).
- 10 rodents had brown fur with short tails (bbll).
- 3 rodents had black fur with short tails (B_ll).

From this information, we can begin to analyze the possible genotypes of the parents in the cross that produced the F2 generation.

Let's consider the first observation: 29 rodents had brown fur with long tails (bbLL).
Since all the offspring have brown fur (bb), one or both of the parents must have had a brown fur allele. Let's assume both parents carried the allele for brown fur.

Considering the second observation: 9 rodents had black fur with long tails (B_L_).
This means that at least one of the parents must have carried the black fur allele (B). Since brown fur is recessive, it is possible for one or both parents to carry it without expressing it.

Considering the third observation: 10 rodents had brown fur with short tails (bbll).
This confirms that at least one of the parents carries the allele for short tails (ll).

Considering the fourth observation: 3 rodents had black fur with short tails (B_ll).
This confirms that one of the parents carries the allele for black fur (B), and the other parent carries the allele for short tails (ll).

Now, let's create a genetic diagram to represent the possible genotypes of the parents:

B______B
| |
b___|____b |
| |
L____L

The possible genotypes for one parent could be BbLL, and the other parent could be Bbll.

In this case, the genotypes for F1 offspring would be BbLL, BbLl, Bbll, and Bbll. During interbreeding of the F1 rodents, these combinations would result in the F2 generation observed:
- 29 rodents with bbLL (brown fur, long tails)
- 9 rodents with B_L_ (black fur, long tails)
- 10 rodents with bbll (brown fur, short tails)
- 3 rodents with B_ll (black fur, short tails)

The genetic diagram and reasoning illustrated above show how the patterns of inheritance can explain the observed results in the F2 generation of the rodents.