A useful introductory book for many issues relating to Turing Machines is "The New Turing Omnibus" by A. K Dewdney (pub. W.H. Freeman; 1993). This book has 66 short chapters (typically 4 - 5 pages long) on various aspects of computer science. The Busy Beaver game is covered in Chapter 39.
Further information is also available via the Wikipedia entry for Busy Beaver
Tibor Radó's original paper on Busy Beaver was published in the Bell Systems Technical Journal. The paper is well written and fairly easy to follow. It covers results for 1-state, 2-state and 3-state Busy Beaver Turing machines and it is freely available online here
A demonstration program (written in C), showing the Busy Beaver (BB) game in action, is available here. Make sure that the game you want to play is uncommented in the main program, then compile and execute the program. The length of the simulated tape is adequate for all the pre-programmed examples given but you may want to adapt the program by adding extra games, a longer tape and a data-driven choice of the game to be played.
At the moment ten games are programmed, where the number of cards, n, varies from 1 to 4. Four of these games are the ones that give the best value Σ(n), for that particular choice of n. Other games either finish very quickly, with sub-optimal scores, or demonstrate that many of the games can simply loop for ever.
Remember that the rules of the game simply ask for an n-card BB Turing Machine that prints the maximum possible number of ones, Σ(n), on the tape and then halts. These ones do not have to be consecutive (though they sometimes will be). It's perfectly all right to have interleaved zeroes between the ones, and indeed this sort of behaviour begins to show up in the n = 4 case, In addition to the Σ(n) value the supplied program prints out S(n), the number of state-transitions (`steps') that the BB Turing Machine had to make in printing out Σ(n) ones.
The first four Σ(n) values of 1, 4, 6, 13, for Busy Beaver, are deceptively small but although the exact value of Σ(5) is not yet known, it has been established that it is at least 4098. The best lower bound so far obtained for Σ(6) is 1018267. To hold a string of this number of ones needs far more character cells on the simulated tape than there are particles in the universe. This, in turn, means that a full exploration of Σ(6) may need more computational capacity than exists in the known universe and hence the values for Σ(6) -- and S(6), the exact number of state transitions that the optimal BB machine would make in printing out Σ(6) ones -- can never be known.
Much of the problem in working out Σ values even for n = 3 and n = 4 lies in determining whether some of the candidate BB-TMs, that appear to be looping, really are in a loop or whether, given long enough, they might actually terminate with an even larger Σ(n) value than the best result known so far. . Radó and Liu ruled out more than 40 such problem cases (`holdouts') in working out that Σ(3) = 6. Allen Brady had to work much harder (in 1983) to eliminate 27 `holdouts' when showing that Σ(4) = 23 and S(4) = 107.
It's worth re-stating that values for Σ(n) and S(n) are truly uncomputable by any general formula, because it can be shown that their values increase more quickly than any computable function (including all `superexponential' functions such as Graham's Number and Ackermann)
The reason for this amazing explosion in Σ(n) values is that these BB Turing Machines really can, in principle, compute anything. So, as n increases, you really are analysing every program that could ever be written! Tibor Radó, in his paper referred to above, speculates that he could, using his BB-TM cards as a kind of assembly language, write a 26-card program that works out n! (n factorial). Since BB Turing Machines are binary (i.e. they use only 1s and 0s on the data tape) the idea would be to initialise the tape with 1111 (say) to represent the number 4 in unary notation and the 26-card machine would then be programmed to read in this string of ones and to work out 4!. Its output would be 24 ones written onto the tape.
But just think of all the other possible TMs in the Σ(26) collection and what they might be capable of doing! Then think even further of what the actual value of Σ(26) might be given that Σ(6) seems utterly beyond us. And if these thoughts don't lead to mental meltdown, then just contemplate Σ(g64) or Σ(ack(n,n)) ....