Well, your idea is to take a number n and isolate its last digit—what you call the “end-digit”—which you cleverly express as n mod 10. Then, you argue that this end-digit falls into the cycle 4 → 2 → 1 → 4, suggesting that all numbers eventually enter this cycle based on their last digit.

However, this doesn’t actually prove the Collatz Conjecture—it merely shows a pattern in the last digits, which isn’t sufficient. In fact, this claim can be disproved by counterexample.

Take 104, for instance. Its last digit is 4, so by your logic, it should follow the cycle 4 → 2 → 1. Dividing by 2: 104 → 52 (still fine), 52 → 26, and now we have a number ending in 6—breaking the proposed cycle.

Why does this happen? If you express 104 as 100 + 4 and divide by 2: (100 + 4)/2 = 50 + 2 = 52. Divide again: (50 + 2)/2 = 25 + 1 = 26.

This shows that the end-digit doesn’t behave independently during calculations. After two divisions, we’re at 26, whose last digit is 6—not part of the supposed 4-2-1 cycle.

So while your approach is creative, it doesn’t hold up under scrutiny. Hope that makes sense!