Collatz conjecture Facts for Kids

German mathematician Lothar Collatz

The Collatz conjecture is a conjecture (an idea which many people think is likely) in mathematics. It is named after Lothar Collatz. He first proposed it in 1937. It is about what happens when something is done repeatedly (over and over) starting at some number n:

If n is even (divisible by two), n is halved (divide by two = take its half).
If n is odd (not divisible by two), n is changed to $3n+1$ .

The conjecture states that n will always reach one. Here is an example sequence:

9
28 (9 is odd, so we triple it and add one)
14 (28 is even; 14 is half of 28)
7 (14 is even, 7 is its half)
22 ( $22 = 3 \times 7 + 1$ )
11
34
17
52
26
13
40
20
10
5
16 (16 is a power of two, so it will lead to 1, halving each time)
8
4
2
1 (after 1 comes 4, 2, 1, 4, 2, 1, etc.)

(oeis:A033479)

Images for kids

Histogram of total stopping times for the numbers 1 to 108. Total stopping time is on the $x$ axis, frequency on the $y$ axis.
Histogram of total stopping times for the numbers 1 to 109. Total stopping time is on the $x$ axis, frequency on the $y$ axis.
Iteration time for inputs of 2 to 107.
Total stopping time of numbers up to 250, 1000, 4000, 20000, 100000, 500000
Cobweb plot of the orbit 10-5-8-4-2-1-2-1-2-1-etc. in the real extension of the Collatz map (optimized by replacing " $3 n + 1$ " with " $(3 n + 1)/2$ ")
Collatz map fractal in a neighbourhood of the real line
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.

Collatz conjecture facts for kids

Images for kids

See also