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The World Champion largest number, listed in the latest Guinness Book of Records, is an upper bound, derived by R. L. Graham, from a problem in a part of combinatorics called Ramsey theory. Graham’s number cannot be expressed using the conventional notation of powers, and powers of powers. If all the material in the universe were turned into pen and ink it would not be enough to write the number down. Consequently, this special notation, devised by Donald Knuth, is necessary. 3^3 means ‘3 cubed’, as it often does in computer printouts. 3^^3 means 3^(3^3), or 3^27, which is already...

Leading banks and investment funds have been foundering, because of bad debts and lack of trust; and other, less well-known kinds of fiscal chaos are also on the horizon. For example, due to an unfixable security flaw in the way funds are now transferred electronically, worldwide, it is no longer safe to write personal checks. A criminal who sees the numbers that are printed at the bottom of any check that you write can use that information to withdraw all the money from your account. He or she can do this in various ways, without even knowing your name ---...