Collatz Conjecture
Start with any positive integer. Halve it if it is even, otherwise triple it and add one. Does every starting value eventually reach 1?
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Suggested experiments
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Follow one orbit runnable above
Enter a starting value and see its steps, its maximum and the sequence. The sample from the problem statement: 27 takes 111 steps and reaches 9232.
Establishes: That this one starting value reaches 1, in the number of steps shown.
Does not establish: Anything about other starting values. It is not a proof of the conjecture.
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Stopping-time records up to N runnable above
Check every starting value from 1 to N (at most 200000) and list each value whose stopping time beats every smaller one. Compare the list with OEIS A006877.
Establishes: That every starting value from 1 to N reaches 1, and which values set records.
Does not establish: Anything beyond N. A finite range is evidence, never a proof.
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Look for other cycles
Write a search for cycles of the rule other than 4 → 2 → 1 within a bounded range, and record exactly which range you covered.
Establishes: Only that no other cycle exists in the range you searched.
Does not establish: That no other cycle exists at all.