File talk:Collatz-tree, depth=20.svg

Shades of grey for number of binary ones?

Suggestion: Fill the ellipses with a shade of grey based on number of binary ones

• light grey: 1, 2, 4, 8, ... all with (1)
• intermediate shades of grey: (2) - (9)
• dark grey: for 349525 (10)

What it will show is some order, not the very chaotic "hailstone sequence" shown by other graphs:

• most numbers flow from many (dark grey) via intermediate shades towards 1 (light grey) binary one.
• giant leaps are exceptional, e.g. 349525 (10) to 1048576 (1)
• some sequences stay within the same shade of grey for some time, e.g. 8832, 4416, 2208, 1104, 552, 276, 138, 69, 208 104, 52, 26, 13, all (3).
• reverse direction is possible, but rare, such as 35 (3) -> 106 (4)

The graph as a whole will illustrate that the Collatz conjecture is probably true, yet not proven. Uwappa (留言) 13:25, 25 June 2024 (UTC)