On the page dealing with Hailstone Signature Analysis mention was made of the fact that as the hailstone process operates on a number, it will reduce the length of that number by roughly one digit for each group of 8 Signature Segments which it executes on that number. For a single number, this is very much an approximation, but the Rule-of-8 demonstration program discussed here allows you to execute the hailstone process on a large group of numbers all of the same length. When the results are averaged over a group of some thousands of numbers, a quite remarkable result emerges as you will soon see. The program is called Rule-of-8 and under default conditions it applies the hailstone process to each of 1,000 thirty digit numbers, and as it does so it accumulates the total number of Signature Segments that were required to reduce all of these numbers to the final value of one. As a final step, it divides this total by 1,000 to determine the average number of Signature Segments required per number. And here's the payoff! That average always seems to be in the 230s or the 240s. That is to say, very close to 8 times the number of digits in the numbers being processed. Using a greater quantity of numbers (say 10,000) will generally yield an average of 238 or 239! For this graphic the Digits per Number was set to 30 and How many Numbers was set to 100,000.
This graphic is in two distinct parts. The top part is a bar graph of the data produced by the batch process, while the bottom part, labeled Hailstone Results, consists of some explanatory notes pertaining to the graph. The following notes will try to make clear just what this graphic is saying:-
Note that the number of Signature Segments is 240 which is the number predicted by the Rule of 8 for this number. That is not an accident ... it's the reason why this number was selected. The Signature Segment Profile tells the story of a well behaved number in that the count value of each segment type is less than that of the previous segment type. Also, for the first few segments at least, the count decreases by a factor close to 2, which is a tendency suggested by probability considerations. Here are the results for the number 257409554728512406177356935231:-
The most outstanding feature here is the high number of OE segments in the Signature. Now an OE segment is the only type which actually increases the size of the subject number, so not only does it boost the count of Signature Segments, it requires an increased number of other segment types to reduce the subject number to its final value of 1. And here are the results for the number 861331426568766895756166034377:-
Here the number of OE segments is only 27 which is less than 10% of the number in the previous case. This, together with the rather long chains of Es in the last few segments listed, conspires to keep the Signature quite short. However it is a relatively simple matter to design plenty of numbers with much shorter Signatures than this. For example 222577830099727861365828048213 which has 6 Signature Segments:-
Or even 422550200076076467165567735125 which has just 1 Signature Segment
In spite of these highs and lows, if you use a large enough batch of numbers when performing a Rule of 8 operation, (1,000 would do although 10,000 would be better) you will almost invariably find that the Average Number of Segments per Signature turns out to be remarkably close to the value predicted by the Rule of 8.
The following table contains the results of a series of tests performed by the Rule of 8 function. The columns contain the following data:-
Note that in every case, the Average Number of Segments per Signature is very close to the (8 times the Number Length) figure predicted by the Rule of 8, and shown in square brackets.
The Highest Number of Segments per Signature is to be interpreted as the highest number encountered on this run. There will probably be a few higher numbers but these can safely be categorized as outliers.
The Lowest Number of Segments per Signature is not recorded. This is because there will always be at least one number for any given number length which will have a Signature with only one segment.
In the page Hailstone Signature Analysis it was confidently predicted that the Rule of 8 would be closely observed as the Hailstone Process is applied to numbers in general. This table demonstrates very convincingly that the prediction was indeed correct.
The progressive results provided by the Rule of 8 function include a profile of the entire set of Signature Segments generated during a batch processing run. The following table was generated during a typical run using one million 100 digit numbers. The Rule of 8 predicts that the total number of Signature Segments should be 8 x 100 x 1,000,000 = 800,000,000. The figure that is reported by the program is 799,180,608. This adds quite significantly to the confidence we can feel in the validity of the Rule.
Of equal interest are the numbers of each of the Segment types. Note that almost exactly half of all the Segments are of type OE, and that all of the subsequent types progressively decrease by a factor of 2. This justifies the equal probability assumptions made in Hailstone Signature Analysis. It is very comforting to observe this behavior as it adds great support to the Collatz Conjecture. But will it always be the case? A more pertinent question to ask is why should we imagine that circumstances might arise where it fails? How would we even begin to define a set of circumstances which would allow it to fail. I will leave that question for others to ponder.