# Everything leads to “one” number

You get an invitation from an Institute of Mathematics and you are told if you can solve a simple puzzle you can be part of their elite institution.

You have to fill the blank.

$6 \rightarrow 6$,
$7 \rightarrow 11$,
$8 \rightarrow 3$,
$9 \rightarrow 13$,
$10 \rightarrow$ __

Can you get into the institute?

5?

Because...

these are numbers 6-10 in the integer sequence Number of halving steps to reach 1 in '3x+1' problem:

0, 1, 5, 2, 4, 6, 11, 3, 13, 5,...

oeis elaborates: The total number of steps to reach 1 under the modified '3x+1' map: T := n -> n/2 if n is even, n -> (3n+1)/2 if n is odd. [The bold part is probably what the title references.]

• ...I hope I don't get much flak for this, but I barely know what I just wrote. :/ – Walt Aug 20 '17 at 15:23
• That was fast. Your answer is what I was expecting. – Invariance Aug 20 '17 at 17:45