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Exponential Growth

Title text: Karpov's construction of a series of increasingly large rice cookers led to a protracted deadlock, but exponential growth won in the end.

Explanation

This explanation may be incomplete or incorrect: Created by an INFINITELY NESTED SET OF RICE COOKERS - Please change this comment when editing this page. No mention of the title text yet.

Exponential growth is the principle that if you keep multiplying a number by a value larger than 1, you will pretty quickly get very large numbers. Even if you start with 1 and simply double it each time, you'll have a 10-digit number after about 30 iterations.

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 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 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 may involve waiting a day between each placing of grains, delaying the unravelling and subsequent outcome of the story). However, by the 20th iteration, there are over 500,000 grains on the board and the king has to dig deep into his supply to continue to pay his dues. On the 24th the king finds he owes more than 8 million grains. By the 32nd, the king finds himself owing over 2 billion grains 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 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.

Since a chessboard contains 64 squares, the final square would contain 2^63 (approximately 9.2 quintillion) grains. This would be around 600 billion tonnes of wheat (even in modern times, this is more than 750 years of global wheat output). Worse, that's just for the final square – adding up all the squares would require about double that (2^64-1 which is approximately 18.4 quintillion grains).

Instead of this being a (possibly apocryphal) story, Black Hat enacts it literally during a game of chess to annoy his opponent into quitting. Black Hat begins describing the metaphor, only to reveal it wasn't a metaphor at all. Black Hat had been playing actual chess games, and tried to force his opponent to resign by burying the chess pieces in rice, as implied by the multiple large sacks bluntly labelled 'rice' on his side of the chessboard. (This is not the first comic to feature large quantities of rice labelled in this manner – in 1598: Salvage, a gargantuan tank of rice has simply the word 'rice' written on the side in equally gargantuan capital letters.)

Garry Kasparov is a world renowned Russian chess master. He had the highest FIDE chess rating in the world - one of 2851 points - until Magnus Carlsen surpassed that in 2013 by 31 points. The Kasparov gambit is an opening move in chess, a variation of the Sicilian Defense.

In 1984-85 Garry Kasparov played Anatoly Karpov in a 5-month-long 48-game championship tournament which was abandoned. In the 1984-85 match Kasparov was losing 4-0 with 6 wins being required to win. Kasparov proceeded to draw 35 times before the match was abandoned.

In a 1985 rematch, Kasparov defeated Karpov for the world championship title, which he retained in their next rematch in 1986.

There are several articles in the International Chess Federation (FIDE)'s Laws of Chess that might prevent Black Hat from winning in this way:

• 7.3 "If a player displaces one or more pieces, he shall re-establish the correct position (...). The arbiter may penalise the player who displaced the pieces."
• 12.1 "The players shall take no action that will bring the game of chess into disrepute."
• 12.6 "It is forbidden to distract or annoy the opponent in any manner whatsoever. (...)"

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 tonnes, 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 of each subsequent row

• • 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)

Example on chessboard (SVG diagram)

Transcript

This transcript is incomplete. Please help editing it! Thanks.

[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 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.

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