In articles which discuss the hailstone problem, you will often find comments to the effect that some numbers should continually increase rather than eventually decrease to 1, based on the fact that odd numbers are multiplied by 3, but even numbers are divided only by 2. Viewing the problem in that way is simplistic and misleading. It is better to break the Signature up into units which I will refer to as Signature Segments, where a segment is defined as any portion of a Signature which begins with O and continues with all of the letters up to but not including the next O. Signature Segments figure prominently in the discussion relating to The Hailstone Tree Without fail, all segments begin with OE, which implies:-
It will terminate at OE or extend beyond OE, with equal probability of 1/2. If it extends beyond OE
It will terminate at OEE or extend beyond OEE, with equal probability of 1/4. If it extends beyond OEE
It will terminate at OEEE or extend beyond OEEE, with equal probability of 1/8. Ans so on... Clearly, the probability decreases by a factor of 2 for each E added to the segment, so longer segments are progressively less likely. However they do have a greater impact on the hailstone process due to the greater number of divisions by 2. The reasoning presented above is captured in tabular form in the following:-.
131054 / 65536 which equals 1.9997. This is the average number of divisions by 2 generated by each Signature Segment. The fact that this number is so close to 2 is significant. In fact, adding additional lines to the table would move it even close to 2. Summing up then, each Signature Segment provides one multiplication by 3 (and an addition of 1) as well as an average of two divisions by 2. Reducing this thought to the simplest possible form implies that on average, calculating one additional Signature Segment for the number multiplies that number by a factor of 3/4. A very interesting circumstance arises when we calculate a series of 8 consecutive Signature Segments. On each of the eight occasions the subject number will be multiplied by a factor which, in the long run, will average out at 3/4. What actually happens is encapsulated in the following mathematical statement.
38 / 48 = .10011In short it gives us a division by a number very close to 10. This in turn translates to a reduction in the size of the number being processed of one digit. To a first approximation then, an n digit number will generate n*8 Signature Segments on its journey to the expected concluding value of 1. This is the basis for what I call The Rule of 8 which you will meet later in this tutorial. I believe you will be pleasantly surprised at how closely numbers right across the vast number spectrum obey this rule when you get to the subject matter considered at Introduction to the Hailstone Program and at The Hailstone Rule-of-8 Demonstration This Rule of 8 is only an approximation (although a remarkably stable one), and as a result small departures from it will be caused by the following:-