Matt's Solitaire Game
Scientific American did an article on this many years ago. (Something like 10-15 years maybe.) The conjecture is that this works for every positive integer. It's been proven for every number less than maybe 1000 or so, except for one, which is - I can't remember. 192, 196, something like that.
And before anyone replies, I know what I just said is inconsistent. If 196 or 192 didn't work, neither would 98 or 96, and I just said it had been proven to work for all other numbers. :-) But hey, whaddya want, it's been a decade or so.
FWIW, 192 collapses to 1 in only 13 steps. 196 takes quite a bit longer - 39 steps.
This is often called the Collatz problem, Kakutani's problem, Ulam's problem, or even the Syracuse algorithm. A cursory search on Google! comes up with a page full of links http://www.geometry.net/Theorems_and_Conjectures/Collatz_Problem.htm
Fog Creek Home