Collatz conjecture
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The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integ
The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×107 (for x = 9663)
The number of iterations it takes to get to one for the first 100 million numbers.
A Collatz fractal centered at the origin, with real parts from –5 to 5.
Julia set of the exponential interpolation.
Hail
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Hail is a form of solid atmospheric precipitation. It is distinct from ice pellets, though the two are often confused. It consists of balls or irregular lumps of ice, each of which is called a hailsto
Collatz conjecture
…to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers. Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." Jeffrey Lagarias…
A large hailstone, about 6 cm (2.4 in) in diameter
Medium size hailstone fallen at Basuki Bihari village on 29 April 2026.
Hail shaft
Severe thunderstorms containing hail can exhibit a characteristic green coloration.