Redstone circuits/Clock

A clock circuit is a redstone circuit which produces a clock signal: a pattern of pulses which repeats itself.

Introduction
Clock generators are devices where the output is toggling on/off constantly. The customary name x-clock is derived from half of the period length, which is also usually the pulse width. For example, a classic 5-clock will produce the sequence  on the output.

Using only redstone torches and wire, it is possible to create clocks as short as a 4-clock, sometimes by exploiting glitches. Using repeaters or pistons allows easy construction of any clock down to 1-clocks, and other devices can also be pressed into service. There are also special circuits called "rapid pulsers", which produce rapid pulses like a 1 tick clock, but inconsistently due to torches burning out. Indeed, torch based rapid pulsers can be too fast for repeaters. Even with repeaters in use, 1-clock signals are difficult to handle in other circuits, as many components and circuits will not respond in timely fashion.

Creating long clocks (more than a few ticks) can be more difficult, as adding repeaters will eventually get unwieldy. However, there are a number of approaches here, which are discussed in a separate section.

Clocks without an explicit toggle can often have one retrofitted, by wiring a lever or other switch to the controlling block of an inverter, or even to a redstone loop. In general, forcing the delay loop high will eventually stop the clock, but the output may not respond until the current pulse has made its way through the loop. Whether the output will be stopped high or low depends on the clock and where in the loop you force it. Another option is to use a lever-controlled piston to open or close one of those loops, using either a solid block to transmit power, or (as of 1.5) a redstone block to supply it.

While it isn't much discussed in the circuit builds below, there is one extra concept which is occasionally important: Phase. The phase of a running clock is the point it has reached in its cycle. For example, at one moment a 5 clock might be 3 ticks into its ON phase, 4 ticks later, it will be 2 ticks into its OFF phase. A long-period clock might be noted as 2 minutes past the start of its ON phase. The exact beginning of a cycle depends on the clock, but it is usually the start of either the OFF phase or the ON phase. For most cases, phase doesn't matter very much, in that you just need pulses every 7 ticks or whatever. However, in-game computing circuits are more demanding, and if you're doing a daily clock, you surely care whether the ON phase is day or night!

Rapid pulsar
Redundancy can be used to maintain a 1-clock, even as the torches burn out; the result is the so-called "Rapid Pulsar" (designs X, Y and (vertical) Z). However, the signal may not be consistent.

Device R creates energy in an irregular sequence. It is a variant of the "Rapid Pulsar" design shown above, except that each torch pulses in an irregular pseudo-random pattern as each torch coming on turns the other three (and itself) off. Occasionally torches will burn out for a few seconds (until reset by a block update), during which time other torches blink. As of version 1.5.1, this is likely to favor one pair of torches, such as the east and west torches, which will blink while the others stay dark. Output can be taken anywhere on the circuit.

Torch loop
The basic torch pulser is the oldest clock circuit in Minecraft, simply an odd number of inverters (NOT gates) joined in a loop. The design has been mostly replaced by repeaters, but still works. Design A shows a 5-clock, which is the shortest clock that can easily be made this way. Its pulse length can be extended by adding pairs of torches and/or repeaters. Repeaters can be added into the loop, or can replace any pair of inverters Adding repeaters also allows even-numbered clocks such as a 10-clock. The total interval will be "NOT gate count"+"repeater total delay".

Even torch based 5-clocks can be made more compact, as with designs B and C. However, these have fewer places where repeaters can be inserted without using more space. Using this method, 1-clocks and 3-clocks are possible, but these will be unstable and erratic as the torches will regularly "burn out". As with the basic clock, the compact clocks can be extended by making the chain of inverters longer, or with repeaters. A 5-clock can also be made vertical, as in G

Design D uses a different method to produce a 4-clock. (A 4-clock is the fastest clock of this sort which will not overload the torches.)

Design E may be obsolete as of version 1.7.  By making use of the North/South Quirk, it was possible to produce a more compact 4-clock with a regular on/off pulse width, as seen in design E. This design uses five torches, but if the stacked torches are pointed north-south, it has a pulse width of 4 ticks.

Repeater clock
A complete clock generator can be simplified to a single block, one Redstone torch and from one to any number of repeaters chained together, as shown in design A. The repeater delay must be at least two ticks, or the torch will burn out.

Clocks also can be made with repeaters only, as in design B. There are no torches to burn out, so they can run at very high speed, such as the 1-clock design C. However, unlike torch-based clocks, these do not start running on their own and must be triggered by hand or with a pulse generator. One way to hand-start a torchless clock is to place a temporary torch which will immediately be turned off by a (previously-placed) power source. The torch will flash on for 1 tick before it realizes it's powered, which is enough to start most fast clocks.

Clock F is built from 2 modified pulse limiters (as shown) in series, and allows for not only an off-switch, but adjustment of how much of the cycle is spent "on". A rising edge on the toggle line will turn the clock on or off. Caveats:
 * The first pulse limiter needs a minimum total of 5 ticks delay between the repeaters, to reliably start the clock.
 * For the second limiter, altering the delays will affect both the output's delay, and how much of the cycle it spends "on". You generally need a total of 10 ticks delay among the 3 repeaters, or the clock will stop.
 * A signal received on the output line can also toggle the state of the clock, so the output should be "isolated" (repeater or inverter) from any circuits it connects to.

Torch-repeater clock


Design D is a tiny vertical clock that can output a 3, 4, or 5-tick cycle.

Earliest Known Publication: June 30, 2011

The period will be the repeater's delay plus 1, but the repeater must be set to at least 2 ticks or the torch will burn out. This circuit is formally 1&times;3&times;3, but is most commonly built as a "V" on the ground, and can easily be buried entirely.
 * A lever on, or redstone signal to, any of the four solid blocks can stop the clock. The torch will be forced "off", while the dust will be lit.
 * Output can be taken almost anywhere, with a few exceptions:
 * The blocks "crosswise" from the redstone dust (pistons work, but dust or a repeater is likely to jam the clock).
 * The block under the repeater (a repeater or piston next to it will be out-of-phase, and dust won't light).
 * Output from the dust side will be reverse phase.

Design E is a more general vertical clock. It can be extended indefinitely to add 2 repeaters (up to 8 ticks of delay) for each block of extension. As shown, it has a minimum delay of 5 ticks, but this can be shortened by replacing some of the repeaters with redstone dust. A lever or redstone signal behind the torch will stop the clock with output OFF, once any current ON-phase passes the output. A signal at the other end (e.g., into the output) will also stop the clock, but with the output immediately going high. (A dust-cut piston will also work.)

Comparator clock
Comparators can be used to make fast clocks and slow pulsers.

Subtraction clock

 * Subtraction 1-Clock


 * 2&times;2&times;2 (8 block volume), flat, silent
 * clock output: 1 tick on, 1 tick off


 * A subtraction 1-clock toggles on and off every tick. It uses a redstone comparator in subtraction mode, with the output feeding to the comparator's side input.


 * When the comparator first receives full power, it outputs strength 15 to the block in front of it, which passes the same signal strength to the dust next to it. The signal strength then declines by 1 (to 14) as it moves to the dust next to the comparator. In the next tick, the comparator subtracts 14 from its it 15 input to output only signal strength 1. This is enough to barely power the block and the dust next to the block, but isn't strong enough to reach back to the dust next to the comparator, so on the next tick the comparator subtracts 0 from its input and the cycle starts again.


 * Only the redstone dust next to the comparator will actually toggle between on and off -- the comparator, the block in front of it, and the dust next to the block only toggle between signal strength 15 and 1. Add additional dust lines to these points to take output from them and allow the signal strength to decline to at least 14 and 0.


 * A subtraction clock doesn't require full power for input -- it will work even with an input strength as small as 2.


 * Variations: You can use any full container as the "input" if a power source would be inconvenient in that location (such as right next to the output).


 * Earliest Known Publication: 9 February 2013.


 * Subtraction N-Clock


 * 2×3×2 (12 block volume), flat, silent
 * clock output: 2-5 ticks on, 2-5 ticks off


 * With the repeater set to a 1-tick delay, this is a 2-clock (2 ticks on, 2 ticks off). Increase the repeater delay to slow the clock down, or even add additional repeaters. If the input strength is higher than 1, the block behind the repeater can be replaced with redstone dust; if higher than 2, the block in front of the comparator can also be replaced with redstone dust. Output can be taken from anywhere.

Fader pulser
A fader pulser is useful for making small clocks with periods less than 15 seconds (for longer periods, hopper clocks can be smaller), but they are difficult to adjust to a precise period. They use a fader circuit (aka "fader loop" – a comparator loop where the signal strength declines with every pass through the loop because it travels through at least one length of two or more redstone dust), renewed by a redstone torch every time it fades out.


 * Fader 9-Pulser


 * 1&times;4&times;4, 1-wide, silent
 * clock output: 1 tick on, 8 ticks off


 * When the input turns off, the redstone torch initially "charges" the fader loop at signal strength 15. There's only one comparator in the loop so each cycle through the loop takes only 1 tick, and the signal strength declines by 2 each time through the loop, so the fader loop will stay charged for 8 ticks. The redstone torch then turns on for only one tick because it short-circuits itself (the torch won't burn-out because it's held off most of the time by the fader circuit).


 * Fader 29-Pulser


 * 2&times;4&times;2, flat, silent
 * clock output: 2 ticks on, 27 ticks off


 * When the input turns off, the redstone torch initially "charges" the fader loop at signal strength 14 at the dust next to the block (the signal strength declined by 1 getting there from the torch). There are two comparators in the loop so each cycle takes 2 ticks, and the signal strength declines by 1 each time through the loop, so the fader loop will stay charged for 28 ticks. One tick later, the redstone torch turns back on, re-powering the fader loop (it stays on for 2 ticks so it overlaps the fader loop's on time by one tick).


 * Variations: Add more comparators to increase the clock's period, or run one side of the fader loop above the other to reduce the clock's footprint.

Hopper clock
A hopper clock (aka "hopper timer") uses the movement of items in hoppers to create a clock signal.

Single-item hopper clock
A single-item hopper clock simply moves a single item in a loop of hoppers.


 * Hopper-Loop Clock
 * 1×3×2 (6 block volume), 1-wide, flat, silent
 * clock output: 4 ticks on, 4 ticks off
 * clock period: 8 ticks


 * This clock just bounces an item back and forth between the two hoppers every 4 ticks. This clock runs while the input is off, and turns its clock signal output off when the input turns on.


 * Technically, the pulse is only 3.5 ticks long (and 4.5 ticks off), but for most purposes this can be treated as a simple 4-clock.


 * Variations: Another comparator can be added to the other hopper to get another clock signal inverted from the other.


 * N-Hopper-Loop Clock
 * 2×(N/2+1)×2 (2×N+4 block volume), flat, silent
 * clock output: 4 ticks on, 4×N-4 ticks off
 * clock period: 4×N ticks


 * An n-hopper-loop clock consists of a loop of hoppers moving a single item around which occasionally powers a comparator output. This clock runs while the input is off, and turns its clock signal output off when the input turns on. The clock period will be N × 0.4 seconds, where N is the number of hoppers.


 * Variations: Other comparators can be added to the other hoppers to get other clock signals out-of-phase with each other.

Multi-item hopper clock
A multi-item hopper clock achieves longer clock periods by using multiple items in the hoppers, and using a latch to keep the items flowing first one way then the other (rather than just bouncing back and forth between two hoppers).

For most of the multi-item hopper clocks, see the Items Required for Useful Clock Periods table (right).


 * Ethonian Hopper Clock
 * 2×6×2 (24 block volume), flat
 * clock period: 8 ticks to 256 seconds (4m16s)


 * When the items finish moving in one direction, the empty hopper's comparator turns off, allowing the associated sticky piston to pull the block of redstone to the other hopper, reversing the direction of item movement. The movement of the block of redstone also updates the other sticky piston (which has been powered for a while) causing it to extend and prevent the first sticky piston from extending again when its comparator turns back on.


 * Powering the hoppers will freeze the clock. Powering one of the blocks or the redstone dust will allow the clock to finish its current cycle before halting.


 * With a single item in the hoppers, the clock has a period of 7.5 ticks (0.75 seconds). Each additional item adds 8 ticks (0.8 seconds) to the clock period.


 * There are a number of useful outputs from this clock:
 * Clock: A regular on/off clock signal can be taken from one position of the block of redstone. The signal will last for half the clock period.
 * Cycle Off-Pulse: Either block faced by a comparator stays powered most of the time, but will turn off for 3.5 ticks every full cycle (but at half-cycle intervals from each other). The power level of the block may vary, so an output repeater may be needed to keep the power level constant.
 * Cycle Pulse: By placing a torch on one of the blocks powered by a comparator, the off-pulse is turned into a regular 3.5-tick on-pulse, once per cycle.
 * Half-Cycle Off-Pulse: By placing two redstone dust alongside or under the positions of the block of redstone, a 1.5-tick off-pulse is generated every half-cycle when the block of redstone moves.


 * Variations: The "1-Wide Compact" version is 1×6×3 (18 block volume). The "1-Wide Tileable" and "1-Wide Upside-Down" versions are both 1×8×3 (24 block volume). [schematics]


 * Earliest known publication: 22 January 2013

Multiplicative hopper-dropper clock
A multiplicative hopper-dropper clock (MHDC) uses a hopper clock to slowly pulse one or more dropper clock multiplier stages to produce very long clock periods (the dropper clock stages "multiply" the clock period of the hopper clock).


 * 2-Stage MHDC
 * 5×6×2 (60 block volume), flat
 * clock period: up to 81.9 hours (3.4 real-life days)


 * The top part is a regular ethonian hopper clock. Once per cycle, the block of redstone will move left and activate both of the droppers in the second stage (the left dropper is powered directly, while the right dropper is activated because it's next to a powered block: the left dropper). The block of redstone in the second stage ensures that only one dropper will actually push an item, forcing the items to move in one direction until the block of redstone moves.


 * If the hoppers contain X items and the droppers contain Y items, the clock period will be X × Y × 1.6 seconds.


 * {| class="wikitable collapsible collapsed"

! colspan="3" | Items Required for Useful Clock Periods ! Period !! Hoppers !! Droppers
 * 10 minutes||125||3
 * 30 minutes||125||9
 * 1 hour||225||10
 * 2 hours||300||15
 * 3 hours||225||30
 * 6 hours||300||45
 * 12 hours||300||90
 * 24 hours||300||180
 * 48 hours||300||360
 * 72 hours||300||540
 * }
 * 6 hours||300||45
 * 12 hours||300||90
 * 24 hours||300||180
 * 48 hours||300||360
 * 72 hours||300||540
 * }
 * 48 hours||300||360
 * 72 hours||300||540
 * }
 * }


 * Adding a third dropper clock multiplier stage increases the maximum clock period to over 10 years. In practice, this may only be needed for clock periods measured in weeks or months (longer than the 2-stage version can provide), generally on servers.

Despawn clock
A despawn clock uses item despawn timing to create a clock signal.

Simply approaching a despawn clock can interfere with its timing, because any player might accidentally pick up the despawning item.


 * Dropper Despawn Clock


 * 3×3×2 (18 block volume)
 * clock output: 5 minutes off, 8-10 ticks on


 * Start the clock by turning off the input. After 5 minutes, the item will despawn (disappear) and the pressure plate will deactivate, allowing the torch to turn on, causing the dropper to eject another item onto the pressure plate.


 * If completely filled with items, the dropper will need to be re-filled every 48 hours, or continually supplied with items from a hopper pipe.


 * Variations: Longer clock periods can be achieved by chaining multiple despawn clocks together, so that each torch triggers the next dropper instead of its own. A dispenser can also be used, instead of a dropper, but is slightly more resource-expensive.


 * Summon Despawn Clock


 * 1×2×2 (4 block volume)
 * clock output: adjustable time off, 1.5 ticks on


 * The command block executes a command to summon an item onto the pressure plate. The exact command will vary, but will look something like this:


 * (summons a stick item next to the command block over the pressure plate)


 * Replace X with a value that determines how long the item should last before despawning: 6,000 - 20 × (for example, 5940 for a 3-second despawn) – this value can be negative for clock periods greater than 5 minutes. Every game tick, this value will increase by 1, and the item will despawn when the value reaches 6,000. Normally, items start at 0 and last 5 minutes (6,000 game ticks = 5 minutes), but setting the item entity's Age changes that.


 * Start the clock by turning off the input.

Piston clock
Pistons can be used to create clocks with a modifiable pulse delay without the use of pulse generators. Pistons can be clocked in a fashion that only leaves the arm extended for the time required to push an adjacent block. However, note that if sticky pistons are regularly used this way (that is, as a 1-clock), they can occasionally "drop" (fail to retract) their block, which will usually stop the clock. (Specifically, if the setup allows for a pulse less than 1 tick long, that will make a sticky piston drop its block. This can be useful, notably for toggles.) Piston clocks in general can be easily turned off or on by a "toggle" input T.

Design A requires only a sticky piston and redstone wire, and is controllable. It runs as long as the toggle line (its power source) is on, and turns off when the toggle line is off. Repeaters can be added to increase its delay. If the repeater is replaced with wire, it can be used as a 1-tick clock, but it is prone to "dropping" its block.

Design B shows how to counter block dropping with an optional, non-sticky, piston. The non sticky piston (the bottom one) is needed for the 1 tick clock as a self repair mechanism. It prevents detaching of the moving block from the sticky piston. If using it only for a 1-tick cycle, the repeater (under the extended piston) can be replaced with redstone wire. The toggle line stops the clock on a high signal.

Design C requires two sticky pistons, and can be easily stopped by just setting one side of the redstone high. The repeaters can be indefinitely extended to make a very long delay clock.

Design D only needs one sticky piston, but at the repeater must be set to 2 or more ticks. If it is set to one tick, the torch will burn out. The output signal can be taken from any part of the circuit. This design can also be controlled; a high input on the toggle line will stop the clock.

The symmetrical design E shows how non-sticky pistons can also "pass around" a block. Output can be taken from any of the outer redstone loops.

Design F is an unusual, stable, 1-tick piston clock. Unlike most repeater-based 1-clocks, its signal is fast enough to make a sticky piston reliably toggle its block, dropping and picking it up on alternate pulses. For The clock to work, the block the piston moves must be placed last. The piston will extend and retract very quickly. The output wire appears to stay off, because it's changing state faster than the game visually updates. However, attaching a piston or other device to the output will show that it is working. The clock can be turned off by a redstone signal (e.g. the lever shown on the block below it) to the piston.

Design G is the simplest design and can be used to create rapid clocks. However, it is not controllable, so the only way to stop such a circuit, without adding additional parts, is to break one component (one redstone wire is recommended). Place a block of Redstone on a sticky piston, then lay down Redstone so that the block powers the piston. Then, once the piston is powered and moves the block, the redstone current will stop, pulling the block back to the original position, which will make the block power the wire again, and so on.

Minecart clock


Minecart clocks are simple, easy to build and modify, but are somewhat unreliable. A minecart clock is made by creating a small track rails with one or more powered and detector rails, arranged so that a minecart can run forever either around the track (A), or back and forth from end to end (B, C). (These need not be sloped -- properly placed powered rails will let a minecart "bounce" off solid blocks -- but you get some extra time as the cart slows down.) A larger vertical track (design C), taken from this video is claimed to produce an exceptionally stable clock. Note that the minecart never quite hits the top of the track.

When running an empty minecart on the loop or back-and-forth, the cart generates redstone signals as it passes over the detector rail(s). Minecart Clocks can be extended or shortened easily by adding and removing track, to adjust the delay between signals. On the flip side, the they are easily disrupted by wandering players or mobs, and a long clock can take a fair bit of space. Also, the exact period is generally not apparent from the design. The need for gold in the booster rails can also be a problem for some players.

Long-period clocks
Creating very long repeater loops can be very expensive. There are several sorts of clocks that are naturally quite long, or can easily be made so:
 * Devices can send item entities through the world: Items flowing on a stream, falling through cobwebs, or just waiting to despawn (that's a 5 minute timer provided by the game).  Droppers or dispensers, and hoppers with comparators, can be quite useful here.
 * A long-period hopper clock is detailed above.
 * The simplest despawn timer is a dropper loaded with disposable items, facing down over a wooden pressure plate. (Additional blocks may be needed to ensure the dropped item lands properly on the plate.)  When triggered by a rising edge, this will produce an ON pulse for 5 minutes, but will not automatically repeat itself.  To upgrade this to a proper clock, two despawn timers can be combined, each set to trigger the other on its own falling edge (end of pulse).  There are two primary hazards with this setup:
 * If the pressure plates are not fully enclosed, the trigger item may fall to one side, stopping the clock.
 * The droppers will eventually run out of items. Two droppers full of (e.g.) seeds will serve for 96 hours, that is 4 days of real time.  If this is insufficient, hoppers and chests can be added to refill the droppers (12 days per chest's worth).
 * Boats and minecarts can also be used with pressure plates or tripwires.
 * Pseudoclocks can even be based on plant growth. For these, timing will not be exact, but they can still be useful for getting occasional signals over long periods.
 * "Factorial stacking" of clocks: Precise clocks (that is, repeater loops) with different periods may be connected to an AND gate in order to generate larger periods with much less expense. One way to make a 60-second (600 ticks) would be to use 150 repeaters set on 4-ticks of delay. Or you could connect two clocks with the periods of 24 and 25 ticks (that's 13 repeaters) to an AND gate.  Note that if the input clocks' ON state is longer than 1 tick, you'll need to filter them with an Edge Detector or Long Pulse Detector, to prevent overlapping on imperfect syncs.  The disadvantages here are:
 * The circuitry can be fairly finicky, and you may need a circuit just to start all the clocks simultaneously.
 * The lengths of the sub-clocks need to be chosen to avoid common factors in their periods. This list of the first few prime numbers will be useful:  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103.  Any one of your clocks can be a power of a different prime, but they can't share factors or they will occasionally "beat" together, causing an extra or missed pulse.
 * A cycle of 1 minecraft day (24000 game ticks, but 12000 redstone ticks) can be produced by stacking clocks of periods 125, 32, and 3. A multiplier (as described below) may be helpful for the longest of these.
 * Then there's the obvious: the Daylight Sensor acts as a clock with a period of one in-game day. The duty cycle can be adjusted by using comparators at different threshold values. Keep in mind that weather may interfere with this, and of course the phase is fixed.

There are also a couple of methods that apply to any clock whatsoever, including irregular pseudoclocks:
 * A T flip-flop can be used to double the period of any clock. This will also convert the pulse to have the same length ON and OFF, if it didn't before.  (Pseudoclocks won't be completely regularized, but they will be smoothed out.)
 * Most powerfully of all, latched repeaters allow production of a general clock multiplier, detailed below. This can be used to multiply the period of any clock, and they can be used in series.

Clock multiplier
This nearly-flat circuit takes a clock input of period  and any pulse length, and outputs as a clock of period , where   is the number of latches used; the output is on for a pulse length of  , and off for the remaining. is limited to 12 or so by redstone signal attenuation; however, the design can simply be repeated to multiply the period again, e.g. a 21-multiplier can be made by chaining a 7-multiplier and a 3-multiplier. This can be continued indefinitely, and unlike factorial stacking there is no restriction on the multipliers.

The build is somewhat tricky: The multiplier loop is in fact a torchless repeater-loop clock. This needs to be started separately, before the latches are engaged. The easiest way to start it is probably to add a temporary "startup circuit" starting 4 blocks from the dust part of the loop: Place a power source, then dust and a block for it to power. Finally place a redstone torch on the block, positioned to power the redstone loop. The torch will flash on for one tick before "realizing" it's powered, and this will start the loop as a clock, which will cycle until the latches are powered. This startup rig can then be removed.

The latches are driven by an edge detector which takes a rising edge and produces an OFF pulse; the pulse length must match the delays of the latched repeaters, so that the multiplier's pulse advances one repeater per edge. The delay/pulse length must also be no longer than the input clock, so it's probably best to keep them both at 1. Note that the delays of the latched repeaters are not actually part of the output period; the latches only count off input edges. The circuit's output is ON while the last repeater is lit and lighting the dust loop.

This circuit need not be fed with a regular clock. With any varying input, it will count N rising edges and output HIGH between the (N-1)th and Nth rising edge.

Variations:
 * A T flip-flop can be used to "normalize" the pulse to half on/half-off, while doubling the output period. Designs L5 from that page is suitable and compact.
 * By separating the latched repeaters with redstone dust (to read their signals individually), this circuit could be generalized into a "state cycler", which can activate a series of other circuits or devices in order, as triggered by input pulses.

Efficiency: An efficient approach to making very long period clocks is to start with a repeater loop of 9 to 16 repeaters (up to 64 ticks), then add multiplier banks with N of 7, 5, and 3 (bigger is more efficient). Doublings should done with T flipflops, as 2 of those are cheaper and perhaps shorter than a 4-multiplier. A couple of notes for the picky:
 * Using two 7-multipliers (&times;49) is slightly more expensive, but shorter, than getting &times;50 with 5&times;5&times;2, or getting &times;48 with 3&times;4&times;4 or 6&times;8;.
 * An 8-multiplier is slightly more expensive, but shorter, than separate 2- and 4-multipliers. However, two of them are both longer and more expensive than three 4-multipliers.

Earliest Known Publication: 22 October 2012.