## Step 1: Define the symbolic representation of the variables
The variables are 'black pens', 'cafeteria chairs', and 'staplers', which can be represented symbolically as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $8x_1 + 9x_2 + 5x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
1. $16x_1 + 18x_2 \geq 29$
2. $16x_1 + 18x_2 + 11x_3 \geq 29$
3. $-10x_1 + 4x_3 \geq 0$
4. $-x_2 + 5x_3 \geq 0$
5. $10x_1 - 4x_2 \geq 0$
6. $16x_1 + 11x_3 \leq 82$
7. $18x_2 + 11x_3 \leq 72$
And the variables $x_1, x_2, x_3$ are integers.

## 4: Represent the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'black pens'), ('x2', 'cafeteria chairs'), ('x3', 'staplers')], 
    'objective_function': '8*x1 + 9*x2 + 5*x3', 
    'constraints': [
        '16*x1 + 18*x2 >= 29',
        '16*x1 + 18*x2 + 11*x3 >= 29',
        '-10*x1 + 4*x3 >= 0',
        '-x2 + 5*x3 >= 0',
        '10*x1 - 4*x2 >= 0',
        '16*x1 + 11*x3 <= 82',
        '18*x2 + 11*x3 <= 72'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='black_pens', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='cafeteria_chairs', vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name='staplers', vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(8*x1 + 9*x2 + 5*x3, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(16*x1 + 18*x2 >= 29)
    model.addConstr(16*x1 + 18*x2 + 11*x3 >= 29)
    model.addConstr(-10*x1 + 4*x3 >= 0)
    model.addConstr(-x2 + 5*x3 >= 0)
    model.addConstr(10*x1 - 4*x2 >= 0)
    model.addConstr(16*x1 + 11*x3 <= 82)
    model.addConstr(18*x2 + 11*x3 <= 72)

    # Update the model
    model.update()

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Black pens: ', x1.varValue)
        print('Cafeteria chairs: ', x2.varValue)
        print('Staplers: ', x3.varValue)
    else:
        print('The model is infeasible')

optimize_problem()
```