Perhaps my point was obscured by my vitriol. Let me try again without the ad hominem.
The upshot of the article at ldolphin.org is that 1 Kings 7:23 predicts a value of 333/106 for pi. The author arrives at this value by extracting the numbers 3, 106, 111, and combining them as 3*111/106. Surely you can see how this might seem contrived to a skeptic. Since these numbers are relatively small, it should not be difficult to find them in a given text. A numerologist of superior skill could produce the ratio 355/113 from the same text.
Now, 333/106 is a pretty good approximation to pi, but it is not remarkable. The reason that it is close to pi is that it is a so-called "continued fraction convergent" to pi. (See this page for an introduction to continued fractions.) The first few convergents to pi are 3, 22/7, 333/106, 355/113, and 103993/33102. Every irrational number can be represented by a continued fraction, and the convergents are used to find rational approximations. (The approximation 355/113 was known to the Chinese in the 5th century AD)
By the way, I think it is silly to call 1 Kings 7:23 a biblical contradiction. The Bible is not an engineering manual, and surely the measurements given are accurate enough for their purpose. On the other hand, I see no difference between numerology and Alex Chiu. (oops, that just slipped out)
Numerology can be used to prove anything whatsoever. The fact that some idiot or liar
can manufacture the ratio 333/106 from a random biblical verse is not particularly surprising or compelling. Also, one wonders why God in his infinite wisdom didn't encode 355/113, which is a far better approximation to pi.
Incidentally, the website you mentioned (ldolphin.org) is a goldmine for skeptics who wish to discredit Christianity. It is filled with some of the most credulous pseudoscientific bilge that I have ever encountered. He makes Art Bell look like James Randi.
It is not too hard to prove that the Hamming code strategy is optimal. I will assume that the
players adopt a deterministic strategy.
Let N be the number of players. There are 2^N possible games, corresponding to the 2^N ways of assigning hats to the players. Let W be the number of games that they win, and let L be the number of games that they lose. (W + L = 2^N)
Let G be the total number of correct guesses made by the players. Since the players have no information about their own hats, G is also the total number of incorrect guesses.
Since winning requires at least one correct guess, G >= W. On the other hand, G <= N*L, since there cannot be more than N incorrect guesses in a losing game. Combining these inequalities gives W <= N*L, thus W/(W+L) <= N/(N+1).
This means that if there are N players, then they cannot win more than N/(N+1) of the time. This bound is achieved by the Hamming code strategy when N is one less than a power of two.
Solving chess is not feasible, but it might be possible to solve checkers. The computer program Chinook has won the (human) world checkers championship, in large part because of its immense endgame databases -- it has perfect information for all positions with eight or fewer pieces.
The long-term goal of the Chinook project is to solve checkers. I don't know if this would be a suitable project for distributed.net, but a complete solution to checkers would be very exciting.
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Perhaps my point was obscured by my vitriol. Let me try again without the ad hominem.
The upshot of the article at ldolphin.org is that 1 Kings 7:23 predicts a value of 333/106 for pi. The author arrives at this value by extracting the numbers 3, 106, 111, and combining them as 3*111/106. Surely you can see how this might seem contrived to a skeptic. Since these numbers are relatively small, it should not be difficult to find them in a given text. A numerologist of superior skill could produce the ratio 355/113 from the same text.
Now, 333/106 is a pretty good approximation to pi, but it is not remarkable. The reason that it is close to pi is that it is a so-called "continued fraction convergent" to pi. (See this page for an introduction to continued fractions.) The first few convergents to pi are 3, 22/7, 333/106, 355/113, and 103993/33102. Every irrational number can be represented by a continued fraction, and the convergents are used to find rational approximations. (The approximation 355/113 was known to the Chinese in the 5th century AD)
By the way, I think it is silly to call 1 Kings 7:23 a biblical contradiction. The Bible is not an engineering manual, and surely the measurements given are accurate enough for their purpose. On the other hand, I see no difference between numerology and Alex Chiu. (oops, that just slipped out)
Numerology can be used to prove anything whatsoever. The fact that some idiot or liar can manufacture the ratio 333/106 from a random biblical verse is not particularly surprising or compelling. Also, one wonders why God in his infinite wisdom didn't encode 355/113, which is a far better approximation to pi.
Incidentally, the website you mentioned (ldolphin.org) is a goldmine for skeptics who wish to discredit Christianity. It is filled with some of the most credulous pseudoscientific bilge that I have ever encountered. He makes Art Bell look like James Randi.
It is not too hard to prove that the Hamming code strategy is optimal. I will assume that the
players adopt a deterministic strategy.
Let N be the number of players. There are 2^N possible games, corresponding to the 2^N ways of assigning hats to the players. Let W be the number of games that they win, and let L be the number of games that they lose. (W + L = 2^N)
Let G be the total number of correct guesses made by the players. Since the players have no information about their own hats, G is also the total number of incorrect guesses.
Since winning requires at least one correct guess, G >= W. On the other hand, G <= N*L, since there cannot be more than N incorrect guesses in a losing game. Combining these inequalities gives W <= N*L, thus W/(W+L) <= N/(N+1).
This means that if there are N players, then they cannot win more than N/(N+1) of the time. This bound is achieved by the Hamming code strategy when N is one less than a power of two.
From the ACS web site: "ACS was chartered by the U.S. Congress in 1876 and is the world's largest scientific society with nearly 159,000 members."
Solving chess is not feasible, but it might be possible to solve checkers. The computer program Chinook has won the (human) world checkers championship, in large part because of its immense endgame databases -- it has perfect information for all positions with eight or fewer pieces.
The long-term goal of the Chinook project is to solve checkers. I don't know if this would be a suitable project for distributed.net, but a complete solution to checkers would be very exciting.
Chinook's web site is http://www.cs.ualberta.ca/~chinook.