## Step 1: Define the symbolic representation of the problem

Let's denote:
- $x_1$ as the number of easy puzzles solved
- $x_2$ as the number of hard puzzles solved

The objective is to maximize points, with easy puzzles worth 5 points and hard puzzles worth 8 points. Therefore, the objective function can be represented as $5x_1 + 8x_2$.

## Step 2: List the constraints

1. Solve at least 3 easy puzzles: $x_1 \geq 3$
2. Solve at least 1 hard puzzle: $x_2 \geq 1$
3. Solve at most 10 easy puzzles: $x_1 \leq 10$
4. Solve at most 5 hard puzzles: $x_2 \leq 5$
5. Solve a maximum of 10 puzzles in total: $x_1 + x_2 \leq 10$

## 3: Symbolic representation in JSON format

```json
{
    'sym_variables': [('x1', 'easy puzzles'), ('x2', 'hard puzzles')],
    'objective_function': '5*x1 + 8*x2',
    'constraints': [
        'x1 >= 3',
        'x2 >= 1',
        'x1 <= 10',
        'x2 <= 5',
        'x1 + x2 <= 10'
    ]
}
```

## 4: Convert the problem into Gurobi code

To solve this linear programming problem using Gurobi, we will use the Gurobi Python API.

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=0, ub=10, vtype=gurobi.GRB.INTEGER, name="easy_puzzles")
    x2 = model.addVar(lb=0, ub=5, vtype=gurobi.GRB.INTEGER, name="hard_puzzles")

    # Objective function: maximize 5*x1 + 8*x2
    model.setObjective(5*x1 + 8*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x1 >= 3, name="min_easy")
    model.addConstr(x2 >= 1, name="min_hard")
    model.addConstr(x1 <= 10, name="max_easy")
    model.addConstr(x2 <= 5, name="max_hard")
    model.addConstr(x1 + x2 <= 10, name="total_max")

    # Optimize model
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution:")
        print(f"Easy puzzles: {x1.varValue}")
        print(f"Hard puzzles: {x2.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found")

solve_puzzle_problem()
```