Editing 2936: Exponential Growth

{{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==
{{incomplete|Created by an INFINITELY NESTED SET OF RICE COOKERS - Please change this comment when editing this page.}}
In this strip Black Hat begins by demonstrating the {{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. 

This principle is often illustrated using a story that generally follows the narrative of a king of India (or elsewhere) wishing to thank a man for creating the game of {{w|chess}}, or perhaps some other chess-related service, and asked him to name his own reward. The man asks for a single grain of wheat (or, in some versions, rice) to be placed on the first square of a chessboard, and then for each subsequent square adding twice as many grains as the one before, until {{w|Wheat and chessboard problem|all 64 squares are filled}}. The king grants his strange request and immediately orders one wheat grain to be placed on the board, imagining this to be a trivial gift compared to the vast riches he had expected to be asked for. For the second square two more pieces are placed, and the square after has four pieces (the tale usually involves waiting a day between each placing of grains, delaying the unravelling and subsequent outcome of the story). However, by the 18th iteration, there are over 500,000 grains on the board in total and the king has to dig deep into his supply to continue to pay his dues. On the 23rd the king finds he owes more than 8 million additional grains. By the 32nd, the king finds himself owing over 4 billion grains (assuming he has come up with the almost equal amount summed up in the 31 previous iterations) and has to give up, realising the essential impossibility of the task.
 		 	
In some versions of the story, the man is executed for embarrassing the king/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 (with rice, rather than wheat), 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 (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 Black Hat, he likely simply doesn't care, and counts it as a win when his opponent stomps off out of annoyance. 

{{w|Garry Kasparov}} and {{w|Antoly 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 an opening move in chess. The title text implies that Kasparov actually tried this 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, 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 his opponent has nothing to resign from - indeed his king still appears to be standing as he walks away.

==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==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[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. 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:Food]]
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