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... There is clearly one point with period 1. There are four with period 2, but we need to include the point of period 1 here, as that returns to itself after being transformed twice, so that makes five points. What about period three? Nine points plus the period 1 point again; 10 points. Period 4 gives ...
... There is clearly one point with period 1. There are four with period 2, but we need to include the point of period 1 here, as that returns to itself after being transformed twice, so that makes five points. What about period three? Nine points plus the period 1 point again; 10 points. Period 4 gives ...
An investigation in the Hailstone function
... The problem of calculating an approximation of the length of Hailstone sequences is split in two. First we derive a formula for the odd numbers, and derive from this the formula for even numbers. If one looks at the Hailstone function one sees 2 calculations, a multiply with 3 + 1 (call it O for Odd ...
... The problem of calculating an approximation of the length of Hailstone sequences is split in two. First we derive a formula for the odd numbers, and derive from this the formula for even numbers. If one looks at the Hailstone function one sees 2 calculations, a multiply with 3 + 1 (call it O for Odd ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.