Put on long pants and thick oven mitts.
Fill a teakettle with water and place the kettle on a stove burner. Turn the stove burner on high heat and allow the water inside to come to a boil.
Take the teakettle to the brick patio. Pour the boiling water slowly over any weeds growing in the brick paver joints, being careful not to splash the water on surrounding plants or flowers. Thoroughly soak each weed.
Repeat Steps 2 and 3 as many times as it takes to saturate each weed growing on the brick patio. The boiling water will kill the weeds within 24 to 48 hours.
Pour the vinegar into the watering can. Saturate the weeds growing on the brick patio with the vinegar. Pour slowly, as you did with the boiling water, to avoid splashing nearby plants and flowers. The vinegar will begin killing the weeds, down to their root systems, in 24 hours.
Check the progress of the dying weeds after 24 hours. Wait up to 48 hours for them to die completely. Pull the dead and withered weeds from the brick patio, using your hands.
Position paver edge restraints against the sides of the brick patio. Secure the edge restraints tightly against the patio edges by hammering 10-inch steel spikes into the ground behind each restraint.
Spread polymeric sand over the brick patio with a push broom. Polymeric sand has an added binder substance that will create a hardened surface, making it difficult for weeds to grow and survive. Push the polymeric sand into the brick paver joints until they are full and level with the top of the bricks.
Sweep excess sand off the surface of the brick pavers. Wash the surface of the patio with water to activate the polymeric sand's binding agent and remove any remaining surface sand. Follow the instructions on the polymeric sand for complete watering instructions.