## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The watch company produces two types of watches: round and square. The production of these watches is limited by the capacity of the teams making them and the quality checking process.

Let's define the decision variables:
- \(R\): The number of round watches produced per day.
- \(S\): The number of square watches produced per day.

The objective is to maximize profit. Given that the profit per round watch is $1000 and per square watch is $1250, the objective function can be formulated as:

\[ \text{Maximize:} \quad 1000R + 1250S \]

The constraints are as follows:
1. Team A can make at most 5 round watches a day: \( R \leq 5 \)
2. Team B can make at most 6 square watches a day: \( S \leq 6 \)
3. The senior watchmaker can quality check at most 8 watches a day: \( R + S \leq 8 \)
4. Non-negativity constraints: \( R \geq 0, S \geq 0 \) (as the number of watches cannot be negative)

## Gurobi Code

To solve this problem using Gurobi in Python, we can use the following code:

```python
import gurobi

def solve_watch_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the decision variables
    R = model.addVar(lb=0, ub=5, name="Round_Watches")
    S = model.addVar(lb=0, ub=6, name="Square_Watches")

    # Objective function: Maximize profit
    model.setObjective(1000*R + 1250*S, gurobi.GRB.MAXIMIZE)

    # Quality checking constraint
    model.addConstraint(R + S <= 8, name="Quality_Checking")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution: Round Watches = {R.varValue}, Square Watches = {S.varValue}")
        print(f"Max Profit: ${model.objVal}")
    else:
        print("The model is infeasible.")

solve_watch_problem()
```

This code defines the problem in Gurobi, solves it, and prints out the optimal production levels for round and square watches, along with the maximum achievable profit. If the problem is infeasible, it will indicate that instead.