explain xkcd - User contributions [en]

2936: Exponential Growth

2025-08-14T02:36:04Z

123456:

{{comic
| number = 2936
| date = May 22, 2024
| title = Exponential Growth
| image = exponential_growth_2x.png
| imagesize = 545x264px
| noexpand = true
| titletext = Karpov's construction of a series of increasingly large rice cookers led to a protracted deadlock, but exponential growth won in the end.
}}

==Explanation==

In this strip Black Hat begins by demonstrating {{w|exponential growth}}, using a variation of the {{w|wheat and chessboard problem}}, a classic demonstration of this mathematical principle. Exponential growth involves an initial quantity being multiplied by any number greater than one again and again. It can cause small numbers to compound into very large numbers faster than might be intuitive. This principle is important in a number of real life applications, ranging from biological growth to inflation to reaction kinetics.

The earliest versions of this story come from India and involve a man (the inventor of {{w|chess}}, in some tellings), being offered a reward by a king, and asking that a single grain of wheat (rice, in some versions) be placed on the first square of a chessboard, two on the second, and each subsequent square having twice as many grains as the one before. In the story, the king generally laughs off such a reward as being trivial, but soon learns that the reward would be impossible to pay. Since a chessboard contains 64 squares, the final square would contain 2<sup>63</sup> (9,223,372,036,854,775,808) grains. This would be around 600 billion tonnes of wheat (which, even in modern times, would be centuries of global wheat production).

In some versions of the story, the man is executed for embarrassing the king and/or being over-greedy; in others, he's rewarded for his cleverness; in yet others he becomes king himself as a consequence. There are also other versions that [https://www.comedy.co.uk/radio/finnemore_souvenir_programme/episodes/7/5/ subvert the well-known tale] by the king not being so naïve as to fall for the 'trick' played by the creator of the problem.

[[Black Hat]] initially appears to be using this example, to demonstrate a mathematical principle, but actually turns out to be using it to "win" a chess match by covering the chess board in rice until his opponent quits out of frustration. Naturally, despite his claims that it's "nearly impossible to counter", under the International Chess Federation ({{w|FIDE}})'s [https://www.fide.com/FIDE/handbook/LawsOfChess.pdf Laws of Chess], this would be illegal on several levels, as deliberately distracting or annoying your opponent is a violation, as is deliberately displacing the chess pieces. Black Hat, being the chaotic [[classhole]] that he is, likely simply doesn't care, and counts it as a win when his opponent [[Hairy]] stomps off out of annoyance.

{{w|Garry Kasparov}} and {{w|Anatoly Karpov}} are both Russian chess grandmasters and former world champions. The two men famously competed for the world championship in the 1980s. The Kasparov gambit is a famous gambit that Kasparov played multiple times (but not, as Black Hat's is, something that can be played very early in the game). The title text implies that Kasparov actually tried Black Hat's method on Karpov, who attempted to consume all the rice with "increasingly large rice cookers", but eventually couldn't keep up. While this is obviously fictional,{{cn}} it fits with the principle of exponential growth. If exponential growth is unrestricted, it will eventually grow beyond the constraints of anything that could plausibly be built to contain it.

In any case, it appears that in his enthusiasm to enact his scheme, Black Hat has neglected to even set up his own pieces (or they have already been completely buried), never mind wait for the game to commence, so Hairy has nothing to resign from - indeed his king still appears to be standing as he walks away, so he may be only assumed to have resigned/defaulted due to competition rules that cover various circumstances in which one may leave the playing area (but, apparently, nothing too restrictive about bringing in sacks of rice).

Another unusual Kasparov gambit is mentioned in [[3082: Chess Position]].

==Math==

The amount of rice collected on each square of the chess board is listed below. It all sums up to around 400 billion tons (or {{w|tonne}}s, the various distinctions being not so important), taking each grain as weighing approximately 0.02 grams. This is 500 times the annual world production.

The last day, alone, would require 200 billion tons. But the implicit nature of this doubling is that the amount of rice you put on at any stage is exactly equal to the amount of rice already on the board ''plus one extra grain''. So there were around 200 billion tons already, before the last square required a virtually identical additional amount.

* First row:
** a1: 1 grain
** a2: 2 grains
** a3: 4 ...
** a4: 8
** a5: 16
** a6: 32
** a7: 64
** a8: 128
* Second row
** b1: 256
** b2: 512
** b3: 1,024
** b4: 2,048
** b5: 4,096
** b6: 8,192
** b7: 16,384
** b8: 32,768
* First column of third to seventh rows
** c1: 65,536 grains (~ 1 kg)
** d1: 16,777,216 (~ 400 kg)
** e1: 4,294,967,296 (~ 100 tons)
** f1: 1,099,511,627,776 (~ 25,000 tons)
** g1: 281,474,976,710,656 (~ 6 million tons)
* Eighth row, in detail
** h1: 72,057,594,037,927,936 (~ 1.5 billion tons, more than the 2022 world harvest)
** h2: 144,115,188,075,855,872
** h3: 288,230,376,151,711,744
** h4: 576,460,752,303,423,488
** h5: 1,152,921,504,606,846,976
** h6: 2,305,843,009,213,693,952
** h7: 4,611,686,018,427,387,904
** h8: 9,223,372,036,854,775,808 (~ 200 billion tons)
* Total: 18,446,744,073,709,551,615

[https://upload.wikimedia.org/wikipedia/commons/e/e7/Wheat_Chessboard_with_line.svg Example on the chessboard (SVG diagram)]

==Transcript==

:[Black Hat is talking to Cueball standing next to him, arm raised.]
:Black Hat: Exponential growth is very powerful.

:[Closeup on Black Hat. Next to him is an image of the lower left part of a chessboard. The four leftmost squares in the bottom row have grains of rice on them -- one, two, four, and eight grains respectively.]
:Black Hat: A chessboard has 64 squares.
:Black Hat: Say you put one grain of rice on the first square, then two grains on the second, then four, then eight, doubling each time.

:[Black Hat has emptied a bag of rice on a chessboard. There are two additional bags next to him, each labeled "Rice", and a pile of rice already on the table. Some rice has spilled off, and a small pile of rice is growing at Black Hat's feet. A frustrated Hairy is walking away, fists clenched. On Hairy's side of the chessboard there is a white king and pawn.]
:[Caption above panel, representing Black Hat continuing to speak:]
:If you keep this up, your opponent will resign in frustration.
:It's called Kasparov's Grain Gambit. Nearly impossible to counter.

{{comic discussion}}

[[Category:Comics featuring Black Hat]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Hairy]]
[[Category:Chess]]
[[Category:Math]]
[[Category:Food]]
[[Category:Comics featuring real people]]

2948: Electric vs Gas

2025-08-14T02:30:36Z

123456: Grammar

{{comic
| number = 2948
| date = June 19, 2024
| title = Electric vs Gas
| image = electric_vs_gas_2x.png
| imagesize = 284x385px
| noexpand = true
| titletext = An idling gas engine may be annoyingly loud, but that's the price you pay for having WAY less torque available at a standstill.
}}

==Explanation==
{{w|Internal combustion engine|Internal combustion engines}} (ICEs) have long been the most common technology used to propel motor vehicles (usually in the specific form of reciprocating "piston" engines). In US vernacular, the most common vehicle fuel is known as "{{w|gasoline}}", or "gas" for short, leading to these engines being referred to as "gas engines". Gasoline is a product of petroleum refinement, leading to the name "petrol" being used in other dialects.

{{w|Traction motor|Electric motors}} would seem the more well-suited method for propelling a vehicle, and as early as 1885 were an actual form of motor car engine with which the fledgling internal combustion engine had to compete. Despite this early popularity, over most of the 20th century electric motors were sidelined in everyday car design, as supplying the electricity was considered to be impractical for most forms of transportation. Modern forms are rapidly rising in popularity, and [https://www.iea.org/reports/global-ev-outlook-2024/trends-in-electric-cars now constitute 18% of all global vehicle sales]. [[Randall]] is a strong proponent of {{w|electric vehicle}}s (EVs).

In this strip, [[White Hat]] claims to be comparing the pros and cons of electric motors and gas engines. The joke is that every point he makes goes in favor of electric motors. Despite it being posed as a dilemma, it may be very clear which side of the debate White Hat is promoting. On the other hand, it may indicate that one of the things we might consider a pro in electric motors (the instantaneous power now available, exceeding that of many non-electric engines) he would consider a problem — perhaps more accurately, a problem with the ''drivers'' of such vehicles — recklessly using the enhanced capabilities to accelerate to high speeds at all opportunities, whether safe to do so or not.

The strip offers the following points in favor of electric motors:

* "Cleaner and more efficient". ICEs produce and vent harmful combustion products and toxic chemicals, while electric motors produce no emission byproducts at the point of use. The efficiency of both gas and electric motors vary, but the typical ICE vehicle in the US converts around 25% of available energy into motion, while the typical electric vehicle is in the neighborhood of 80%. Even when considering inefficiencies in the source production and transmission and storage and release of energy, battery-driven electric vehicles are generally more efficient than internal combustion propelled vehicles[https://www.factcheck.org/2024/02/electric-vehicles-contribute-fewer-emissions-than-gasoline-powered-cars-over-their-lifetimes/].
** It should be noted that all of this refers to the motors only, and ignores how the fuel and electricity are produced, or the wider environmental impact of the vehicle. Determining the overall environmental advantages of electric vehicles is a much more complex and involved calculation.

* "More powerful". Electric motors are able to deliver a lot of power from a small motor ''if'' an ample energy supply is available, and can do so 'on demand', often far quicker than a fuel-powered engine that has to put its power through a gearbox in order to service a wide range of road velocities, from standstill to the eventual top speed. Due to battery limitations, short or partial runtime use cases (such as dragsters, hand tools, yard tools, toys and electric scooters) net the most benefit from the small size of a high-powered electric motor.

* "Annoyingly loud". ICEs, by their nature, produce significant noise. Despite noise attenuation measures such as mufflers, they contribute significantly to urban noise. Properly designed electric motors are nearly silent (even if the rest of the vehicle is not). In particular, {{w|blowoff valve|turbocharger blowoff valves}} make particular noises that are completely lacking in an all-electric vehicle being driven at a similar performance level. This might legitimately be considered a problem, though, when everyone is used to a rapidly approaching vehicle providing a very noisy warning of its approach. EV makers have sometimes added [https://www.businessinsider.com/electric-car-ev-fake-engine-sound-hyundai-dodge-toyota-2023-7 fake ICE noises] to appeal to older drivers, and in the U.S. and some other countries, EVs are required to have {{w|Electric vehicle warning sounds|warning sounds}} at low speeds for pedestrian safety.
** At highway speeds, the noise of tires against the road is much louder than a properly muffled ICE, so the intrinsic quietness of an EV's motor is close to irrelevant in that context.

* "WAY less {{w|torque}} available at standstill". ICEs need to continually operate within a specific range of rotational speeds for best power and fuel efficiency (although the reciprocating engines used in most motor vehicles are still better than some others, such as {{w|gas turbine}} engines, in this regard), which means that a complex system of {{w|transmission|transmission gearing}} is needed to convert this motion into the specific speeds needed at the wheels. When starting from a standstill, this means that torque must be applied to the wheels relatively gradually to avoid stalling the engine. In addition, when a vehicle is standing still, the motor is typically idling at (very) low speed and must be sped up before it produces significant acceleration. Electric motors, in contrast, generally produce their peak torque when at a standstill. This results in electric vehicles having significantly better acceleration and engine responsiveness. Again, this could cause a legitimate problem with drivers changing from ICE to electric motors, because the new cars accelerate more than the driver is used to and provide different feedback. The audible clues of gear changes, whether from automatic or manual systems, are part and parcel of what many people have grown up with and come to rely on in anticipating what might need paying attention to.

It should be noted that White Hat is deliberately confining his arguments to electric vs gas ''motors'' rather than electric or gas-powered ''vehicles''. Doing so ignores the basic reason why internal combustion vehicles have long been the dominant form of personal, motorized transportation: hydrocarbon fuels are a very dense and fairly easy to handle form of energy ''storage''. Providing electrical power to a moving vehicle requires either that the vehicle remain in contact with a power line (as with an electric train or a tram) or else to carry a high-capacity battery (and the ability to recharge that battery in a reasonable amount of time, while stationary). More popular in the USA is a hybrid system, where a combustion engine provides at least some of the power to an electric motor, which was impractical until comparatively recently. Other methods, such as hydrogen fuel cells (a form of "combustion" that can be used more directly to form electricity), have been proposed, but remain experimental or niche, due to various barriers to adoption.

A more comprehensive comparison would include many more factors, both against and in favor of electric cars.

Issues raised with electric vehicles typically include:
* Higher cost of purchase (primarily due to the cost of batteries and, in the USA, now a 100% tariff on Chinese EVs), although partially offset by lower costs of operation
* Long charging times compared to refilling a gas tank (there are some approaches which mitigate this by operating either very high-powered chargers or a battery swap model, rather than charging in-car, but these are not widely adopted)
** The significance of this depends on individual situations. Most EV drivers charge primarily at night, and many drivers don't mind a charging break on long trips.
* Relatively limited range. This is less of an issue in modern times, as many modern EVs have ranges of ([https://ev-database.org/compare/electric-vehicle-longest-range 200-500 miles per charge] as of 2024, similar to the range of a typical ICE vehicle.
* Shortened range in hot weather and significantly shortened range in cold weather (while all vehicles have this problem, it's more pronounced in EVs compared to ICEs)
* Limited charging infrastructure compared to the prevalence of fuel stations
** Because many drivers can charge at home, fewer public chargers are needed than for gas cars.
* Higher vehicle weight, and resulting higher particulate emissions (from tires, but not brakes, because EVs' use of regenerative braking reduces wear on their traditional brakes)
* Reliance on some mineral and metal extraction industries (e.g., lithium) with capacity that lags the recent increased demands for EVs
* Increased demand on electricity production
* [https://archive.is/6xmJV Lower reliability](data possibly skewed by the [https://community.cartalk.com/t/should-i-still-avoid-the-first-model-year-of-a-new-vehicle/118110 newness of EV models]).

Other real pros of electric cars are also not mentioned:
* Lower total cost of ownership (TCO), due to cheaper fuel and lower maintenance costs. [https://www.edf.org/sites/default/files/2023-0/WSP%20Total%20Cost%20of%20Ownership%20Analysis%20July%202023.pdf]
* Can potentially "fill" them at home or while parked without having to stay with them (partially negates the issue of "long charging times".)
* Lower carbon footprint, and reduced dependence on the fossil fuel industry

Rapidly evolving technologies, government policies, and economic realities are changing the relevance and seriousness of these points over time. As of the publication of this strip, the "pros" of EVs do not seem to be universally convincing, as ICEs remain far more popular than EVs in most countries (EVs constitute a majority of new vehicle sales in only four countries: Norway, Iceland, Sweden and Finland). That said, EVs didn't exist as a viable industry 20 years ago, so the current reality reflects rapid and ongoing growth, suggesting that the advantages of EVs are gaining increasing recognition and understanding.

The degree of adoption is also likely to impact the viability of different vehicle types. Infrastructure in most countries has long been built around an assumption of ICEs, so things like fueling stations and ICE-qualified mechanics have traditionally been widely accessible. As EVs becoming increasingly dominant, this could shift, with EV charging infrastructure becoming easier to find than ICE fueling stations, and ICE mechanics potentially becoming more difficult to find. There's typically a certain level of inertia in the adoption of any new technology.

In the EU, [https://www.europarl.europa.eu/topics/en/article/20221019STO44572/eu-ban-on-sale-of-new-petrol-and-diesel-cars-from-2035-explained the sale of new ICE cars is banned from 2035] in an effort to move to EVs, and other jurisdictions are adopting similar policies.

Especially in the United States, this topic is highly contentious for political, economic, engineering, and {{w|Rolling coal|tribal}} reasons (as a quick look at the edit history of this page will confirm).

==Trivia==
The etymology of "gasoline" (commonly abbreviated to "gas") is disputed: it may refer to the gaseous {{w|state of matter}} as, though gasoline itself is generally liquid, it readily emits volatile vapours. Otherwise, it has been suggested to have derived from "Cazelin"/"Cazelline", originally a lamp-oil, sold by a man called {{w|John Cassell}} in the 19th century and much copied (including by the "Gazeline" brand) as the market and supply of such fuels expanded. In non-American english, the octane form of motor-fuel (i.e. non-diesel and excluding aviation fuel) is called "petrol" (derived from "petroleum", or 'oil of the rock'), with similar "electric vs petrol (vs {{w|Hybrid electric vehicle|hybrid}})" comparisons. The registered brand name "Petrol" (sold as a solvent, before being repurposed for use as vehicle fuel) could not be trademarked as it was already the common generic term for equivalent products. The unrefined crude mineral oil from which many different hydrocarbon products can be refined is still more widely known as petroleum. Vehicle fuel may now be partly or wholly composed of non-fossil-fuels to distance them from some of the traditional arguments against petrol/gas consumption, leave other considerations unchanged but possibly introduce {{w|Biofuel#Issues|further issues}}.

While ICE vehicles have, to date, proven more popular than electric equivalents, worldwide, bicycles have been outselling cars [https://www.bikeradar.com/features/bike-vs-car-the-sales-race since at least WWII].

==Transcript==
:[White Hat, with his palm raised, is talking to Cueball.]
:White Hat: Electric motors and gas engines each have their pros and cons.
:White Hat: On one hand, electric motors are cleaner and more efficient. On the other hand, electric motors are more powerful.
:White Hat: So it's hard to say which is better overall.

{{comic discussion}}

[[Category:Comics featuring White Hat]]
[[Category:Comics featuring Cueball]]
[[Category:Engineering]]
[[Category:Climate change]]