Pi appears in non-circular formulas mainly because of hidden rotational symmetry and because oscillations are mathematically tied to circular motion. Some appearances come from how constants are defined.
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In a random group of just 23 people, the chance that at least two share a birthday is slightly over 50 percent. It seems wrong, but it is true.
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The theorem says four colors suffice to color any map so neighbors differ. Its 1976 proof relied on a computer, which raised doubts about checkability and whether it counts as a proof.
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Imaginary numbers solve equations that real numbers cannot and simplify calculations involving waves, so they are central to electrical engineering, signal processing, radar and physics.
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Yes, by most estimates. Shannon's figure of about 10^120 possible games far exceeds the roughly 10^80 atoms, though the number of positions is much smaller.
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The arithmetic holds: 42 doublings of a 0.1 mm sheet give about 440,000 km, past the Moon. But the fold cannot be done, and a geometric argument says the sheet itself would need to be enormous.
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Conway's Game of Life is a zero-player cellular automaton devised by John Horton Conway in 1970, in which four simple rules applied to a grid of cells produce complex patterns.
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