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This tutorial seeks to teach you how to refresh a redstone signal with absolutely no delay. It involves clever uses of BUD-powered pistons.
The most common instant repeater requires a pulse at least four redstone ticks long. It is made by having a redstone signal powering a block with a sticky piston on top of it, facing the direction of the rest of the signal. This sticky piston has a redstone block on its face. The block under the sticky piston is also attached to a sticky piston, but this new piston is facing the opposite direction and is diagonally below the redstone block such that the redstone block can BUD-power the bottom piston when the top piston is retracted. The rest of the signal is powered when the redstone block is extended forward.
This type of instant repeater works by BUD-powering the bottom piston. When the pulse is received, the top piston starts moving. This updates the bottom piston, causing it to retract at exactly the same time as the top piston is extending, thus depowering the top piston by pulling its power source away. This 0-ticks the top piston, effectively teleporting the redstone block above the bottom piston. The redstone block is now powering the bottom piston, so it extends. The block on its face comes back to its original position, but the start of the redstone line is still powered. This powers the top piston, and retracts the redstone block, depowering the redstone line at the end and BUD-powering the bottom piston again. In real time, it looks like the contraption does an "inchworm." This produces pulses that are two redstone ticks long, which means that they can be chained together.
Alternatively, a dual-edged monostable instant redstone line can be made by placing a slime block above a piston and a redstone block above that, and placing an obsidian block on the side of the slime block. A redstone line on top of the obsidian should be extended to the ground. Any block updates next to the piston will send two redstone pulses down the wire, which allows the lines to be chained together, one every 16 blocks. The powered off redstone in the schematic represents the input, and any change in its state will result in the activation of the BUD.
Any time a redstone signal needs to get somewhere instantly, an instant repeater can come in useful. Here are some common scenarios: