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Bits and Pieces from the NR Trivia Collection
#10: Top Runners' Conference

by Jens Kreutzer

Top Runners' Conference represents one of the most powerful bit-gaining schemes in Netrunner. For an investment of 0 bits and an action, the Runner gains 2 bits each turn for the rest of the game - as long as no run is made, that is. Obviously, this is not a good idea in a stack that does a lot of running early on, but if this resource survives just two Corp turns, it is already on par with Score! Another elegant trick is to combine it with Smith's Pawnshop: If the Runner only runs every second turn and installs a Conference right after running, it is possible to gain 6 bits per action out of it.

In the long term, Top Runners' Conference in multiples beats even Loan from Chiba for a huge bit buildup that is then used up in one fell winning swoop (like The Big Dig or Masochism Rules), often helped along by a misc.for-sale for even more bits. Top Runners' Conference ranks among the most sought-after cards in the game, not only because it is such a powerful and useful rare, but also because players don't want three (like Political Overthrow) or six (like World Domination), but as many as they can get - for this prestige card only shows its true potential if you have a lot of copies in your stack.

The card Top Runners' Conference also lent its name to The Top Runners' Conference (TRC), the official Netrunner Players' Organization, and a stylized version of the cool artwork by James Allen Higgins has been turned into the logo of this newsletter: A sphere connected with and surrounded by a circle of eight other spheres (though on the card, there are actually ten shapes in the circle). The picture apparently shows a conference in Netspace, where ten Runners have gathered around a central spherical object that is either another Runner or perhaps some matter of importance being discussed. The Runners are not depicted as in real life, but as the icons they use when they roam the Net - the virtual form in which they appear to everybody else they encounter there. Among various geometrical and abstract shapes, a rattlesnake and a sphere with the letter M (for Militech?) stand out. One icon in the background is a sphere that seems to have several balls floating around it - maybe an atom or a solar system, or perhaps a Beholder, a monster that is featured in the D&D roleplaying game (the sphere seems to have a single eye and a gaping, grinning mouth, which would fit).

As enjoyable as the card's ability and artwork is its flavor text, which reads: "I have discovered a truly elegant codebreaking routine. Unfortunately, this chip is not large enough to contain it." Jennifer Clarke Wilkes has revealed that she was the author of this text, and that its reference to a certain, very famous mathematical problem was intentional. Apparently, Richard Garfield, who holds a Ph.D. in mathematics, was impressed and amused by this. The problem referred to is known as Fermat's Last Theorem. Pierre de Fermat (1601-1665) was a French mathematician who wrote an annotation into the margin of his copy (now lost) of Bachet's translation of Diophantus' Arithmetika. Translated into modern English and modern terminology from the Latin, his comment amounts to:

[Note: It's difficult to represent this mathematical stuff in HTML. Please read "a$b" as "a to the power of b".]

"a$n + b$n = c$n has no positive integer solutions for a, b and c when n > 2. I have discovered a truly remarkable proof which this margin is too small to contain."

(In the original: "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.")

The following is an excerpt from the Microsoft Encarta, "Fermat's Last Theorem," Microsoft Encarta Online Encyclopedia 2001:

"While studying the work of the ancient Greek mathematician Diophantus, Fermat became interested in the chapter on Pythagorean numbers - that is, the sets of three numbers, a, b, and c, such as 3, 4, and 5, for which the equation

a$2 + b$2 = c$2

is true. He wrote in pencil in the margin, 'I have discovered a truly remarkable proof which this margin is too small to contain.' Fermat added that when the Pythagorean theorem is altered to read

a$n + b$n = c$n,

the new equation cannot be solved in integers for any value of n greater than 2. That is, no set of positive integers a, b, and c can be found to satisfy, for example, the equation

a$3 + b$3 = c$3

or

a$4 + b$4 = c$4.

Fermat's simple theorem turned out to be surprisingly difficult to prove. For more than 350 years, many mathematicians tried to prove Fermat's statement or to disprove it by finding an exception."

Quoted from A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball:

"Except a few isolated papers, Fermat published nothing in his lifetime, and gave no systematic exposition of his methods. Some of the most striking of his results were found after his death on loose sheets of paper or written in the margins of works which he had read and annotated, and are unaccompanied by any proof. It is thus somewhat difficult to estimate the dates and originality of his work. He was constitutionally modest and retiring, and does not seem to have intended his papers to be published. "

Quoted from an article by J. J. O'Connor and E. F. Robertson:

"Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved [for a long time]. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example, over 1,000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993. [...]

"The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i. e. curves of the form y$2 = x$3 + ax + b for constants a and b. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. In 1986, the connection was made between the Shimura-Taniyama-Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrucken, showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.

"Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter-example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England, the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning, Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard, he said, 'I will stop here', and sat down. In fact, Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

"This, however, is not the end of the story. On 4 December 1993, Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However, one problem remained, and Wiles essentially withdrew his claim to have a proof. [...]

"In March 1994, Faltings, writing in Scientific American, said: 'If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.' Weil, also in Scientific American, wrote: 'I believe he has had some good ideas in trying to construct the proof, but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.'

"In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However, they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994, Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties. Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck: 'In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.' On 6 October, Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one."

So, using modern mathematical methods and more than a hundred pages in the process, Fermat's Theorem has finally been proved to be correct (though people without an academic mathematical background probably would have a hard time understanding this proof). However, Fermat couldn't have known all of these modern methods back in around 1630, and it remains a mystery how he could know (or why he thought he knew) that his theorem was true.

(By the way, Classic's agenda Theorem Proof is almost certainly a jab at the voluminous book that had to be written in order to prove Fermat's Last Theorem.)