Repairing a brick walkway isn't a difficult job but be sure to do it correctly the first time.
Step 1: Removing the Brick
One of the most aggravating parts of the repair of a brick sidewalk can be removing that first brick, especially if the walk has been in place for a while and settled. There are a number of reasons a sidewalk may need repair. You may have broken or cracked bricks or bricks that are higher than others and present a tripping hazard. If the later is the case then you will need to remove more than one brick to flatten the walk again to be safe.
The most difficult part is removing that first brick. Slide the point of your masons trowel between the bricks at the short end of the raised brick and wiggle it a bit and lift. You may need to do this several times to get it loose. You may also have to go to the opposite end of the brick and do this same thing to get it to pop out. This may take a bit of time and work. Once the first brick is removed the rest are easy to lift up. Keep these bricks if none are broken or cracked.
Step 2: Determine the Problem
If the bricks were buckling upward then there may be a tree root beneath them. This root will need to be cut and removed so you can flatten the walkway. Use a saw to cut the root as far back as you can to prevent having to do this fix again in the near future.
If the bricks were sinking then its possible that the sand and gravel underneath the bricks wasn't level and tamped down when you put the walkway in. Its also possible erosion has occurred. Fill in the depression with more sand and gravel and tamp down until the area is hard. You can use one of the bricks you removed and the rubber mallet to do this. Don't skip this step.
Step 3: Replacing the Bricks
This sounds a lot easier than it may actually be. For some reason it always seems you have less space when replacing them. Start by putting the first couple of bricks in place and then use your rubber mallet to tighten them together. Tap them on the short ends to force them into place. Don't put sand between the bricks until you are toward the end of the project and can see they all are going to fit with a limit of fuss. Continue to put them in, tapping them in place. When you get to the last brick do the same but don't be surprised if its a tight fit. Use the rubber mallet to hit the brick on the top, flat side to force it into place. You may need to tap the other bricks tighter into place to fit this last brick in. Then put sand over the top of all these bricks, sweep it into place and tap it down.