The hopper is a useful device for controlling/counting/sorting items.
Day counter
This is a very simple machine for counting days. Every time it becomes day, the dispenser is activated and shoots an item into the chest, but you have to be nearby for this to activate, making this fairly useless.
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Item counter
A mechanism that outputs short redstone signal for every item that goes through the dropper. The mechanism slows down the items moving through the dropper and the hopper above to make the outputs comfortable to use for counting mechanisms.
The output of this counter can be used in any counting mechanism and can be counted using the scoreboard command in a command block.
It is possible to change the outputs timing by replacing the bottom comparator with repeater (to make it slower) or redstone dust (to make it faster).
Size: 1×4×5 blocks.
This mechanism made by Xbxp.
Item sorter
Introduction
Most item sorters work using hoppers.
More specifically, a hopper with all five slots occupied by, for example, redstone, will only be able to collect redstone, because there is nowhere for any other items to go.
Connecting the hopper to a comparator will allow the sorter to measure the contents of the hopper. All five slots MUST remain occupied by the material you are sorting for, otherwise other items will be able to pass through. The easiest way to ensure this is to measure for a 2-strength signal from the comparator, which corresponds to 22 items inside - 18 inside the leftmost slot and 1 each in the other four slots (hoppers lose items in the leftmost slot first). You can also sacrifice an item that you will never sort for, like sticks, 18 inside the rightmost slot, 1 each in the other three slots, and the material that you are sorting for, in the leftmost slot. Alternatively, you can use an item that stacks to 16 (like snowballs). Place a stack of the item to be sorted in the first slot of the hopper. Then, place two snowballs in the second slot and one in each of the remaining slots.
If the output double chest fills up, though, the above design will break adjacent cells due to the signal from the hopper exceeding 2. This alternative design avoids that. The upper hopper here will contain 41 of the item being sorted in one slot, with the other four slots having one each of some item that stacks to 64 and will never go through the sorter (an arbitrary item renamed in an anvil may be used for this purpose). This variant is often referred to as the "impulseSV item filter".
Components
There are two components to an item sorter: the item channel and the sorting component(s). Here is an analogy to clarify this: the item channel is like a river, and the sorting components can be imagined as nets that only catch certain items. The river ends in a dump (the junk chest).
The item channel is simply a path that items travel through. It can be a horizontal chain of hoppers, or a water channel. Hoppers underneath the hopper chain or water channel act as the "net" and will extract items.
The sorting components are composed of hoppers. To enable deposition and extraction to be independently toggled, two hoppers must be used: one on top of the other, and the top one pointing sideways. The top hopper should be connected to the comparator, which will power the bottom hopper (which takes items out of the top hopper)
Filters for non-stackable items
A variant of sorters is a filter for non-stackable items. These items can not be separated using the layout described above, because any individual item would completely fill a slot in a hopper. Instead, this filter operates on empty hoppers and detects different strengths of redstone signals as stackable or non-stackable items pass through the input hopper. Stackable items (both stackable to 16 and to 64) generate only signal strength 1 in the redstone comparator, while non-stackable items generate signal strength 3. The filter is constructed in a way that a signal strength of 1 will not reach the redstone torch, which by default powers the redstone dust dot below it. The dust dot powers the full block next to it, from where the redstone repeater receives its signal and powers the full block next to the middle hopper to lock it.
Note that the middle hopper only gets unlocked long enough to pull in a non-stackable item and potentially push out which ever item it previously contained. Place another hopper below it, as shown in the schematic, to prevent individual items from being held in the middle hopper. That bottom hopper can face in any direction.
Separating potions and shulker boxes
These two filters work similarly, in that they consist of a hopper that attempts to push an item into a container, but that container has restrictions on the type of items that can be pushed into it. On a delay, if that push fails, a second hopper will pull the failed item from the input hopper. The potion filter attempts to push items into a brewing stand, which only accepts bottles and brewing ingredients (although, you probably don't want to run ingredients through this filter, because they can't be pulled out), and the shulker box filter attempts to push items into a shulker box, which will not accept other shulker boxes. When the item is rejected by the container, the second hopper becomes unlocked to pull it from the input hopper after a delay of three redstone ticks.
To ensure every item gets an attempt to be pushed into the container, the input rate must be throttled to at most one item every 4 redstone ticks. You could, for example, push items with a dropper on a 4-tick clock.
Auto Smelter/Cooker
Ores and cookable items can be automatically smelted while you're away.
Things you want to be cooked/smelted go into the top chest, Fuel into the lower one. The hopper can put in fuel not only from the side of the furnace but also from the front or behind. The product collects in the bottom hopper or is then redirected into another hopper or another chest.
Note that this method forfeits the XP earned from smelting things. [new with 1.13] Experience will get saved in the furnace until a player picks an item manually.
Hopper Clock
Connect 4 hoppers in a circle (create one pointing at nothing, point a 2nd one at it while crouching, destroy the 1st and use the 2nd as the start of your circle since it's the 1st one pointing the right direction). Put any item that doesn't stack (a wooden axe for example) into any one of the hoppers. Attach a comparator to any one of the hoppers. It will give a signal every time the item goes through that hopper. If you want it to activate more often make a 2 hopper clock. If you want to make it activate less often, place more hoppers.
Slow Clock
Connect two hoppers (let's call them hopper A and B) with their outputs facing each other. Measure their outputs with comparators (comparator A and B). Connect the comparators' outputs to repeaters (repeater A and B), add repeaters perpendicularly to make them lockable (lock A and B). Connect the output of repeater A to hopper B and lock B and the output of repeater B to hopper A and lock A.
To start the clock, throw items into one of the hoppers. The speed of the clock will depend on the amount of items circulating in the system, with the longest duration being 128 seconds in case of 5 stacks of 64 items.
Silo
Sometimes we need a storage way larger than just a single large chest. What we need is a single output container which will be automatically filled up with contents of a large set of other containers preferably hidden behind a wall or a ceiling. We call such system a silo or a mass storage.
The basic form
This is a basic form of silo with 3 large chests. It is 3 blocks wide, 5 blocks high, and is tileable. The actual input of the system is the hopper below "I" so you can push items into it from the side not only from above. The bottommost hopper can also be replaced with a hopper facing any direction, feeding into another hopper.
If you want to use a chest for the output container, it may be advisable to use a trapped chest rather than a normal one. This way item refilling gets suspended while you are accessing the chest so you can return items in case you took more items than you need.
Extended forms
Its true power comes from the extendability. Here are some examples of extended forms. All of these are still tileable.