The simplest impossible problem
A sequence lurks in this world. We never seem to be able to beat it. Seeing all our trials comes down to the pattern '4,2,1,4,2,1'. You guessed it, it's the 3n+1 sequence. So today, i'm gonna put you all on a quest.
Find a positive integer that doesn't reach '4,2,1,4,2,1' in the 3n+1 sequence. Is it possible? Or do all positive integers with a finite number of digits go down to that pattern? Remember, the 3n+1 sequence goes as follows,
If the current number is even, divide it by 2.
If the current number is odd, multiply it by 3 and then add 1 to that new number.
Try to find such a positive integer.
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1 solution
This sequence is known as the hailstone sequence. So far, we have never been able to prove that a number exists that doesn't eventually lead to 1, but neither have we been able to prove that all numbers with a finite number or digits will lead to 1.
The answer is not "not possible," it's just that we don't know if such a number exists.