There's been some recent news on the Busy Beaver problem. [Consider all Turing Machines with C symbols and S states WHICH HALT when started on a blank tape. The problem is to find the longest running machine, and the one which leaves the most non-blank stuff on the tape. These might be different. The general problem is of course unsolvable. C=2 S=4 is known. Proposed by Rado in the 1960s.] The current record for 2 symbols & 6 states is 10^1730 steps and 10^865 non-blank symbols. The "magic" is that the tape has a state variable (in unary?), and a quadratic function like 2x^2+3 is applied to it, halting when some modular condition (like x=3 mod 5) is reached. The record 2&5 machine is similar, but the iterated function is only linear, such as 8x+3. The step count is 47 million, and it prints 4098 non-blanks. Marxen has posted his 1990 paper with these results. The record for 2 symbols & 5 states has stood for 15 years, but there must be a small residuum of unknown-outcome machines that run for a long time and seem to be chaotic. It's not clear if there's been any progress here, even toward reducing the number of unknowns. The record 3&3 machines produce 36K non-blanks, and (a different machine) run 544M steps (Myron Souris). See Heiner Marxen's page for details. http://www.drb.insel.de/~heiner/BB/ Perhaps the 2&7 winner will iterate 2^x+3. But if the 2&5s can't be polished off, then the larger ones are beyond hope. Rich rcs@cs.arizona.edu