The analysis carried out at Hailstone Signature Analysis suggested that the hailstone Signature of a typical number can be expected to contain approximately eight times as many Signature Segments as the number of digits in the number. On the other hand it was quite apparent from the graph at  Hailstone Rule of 8 that quite significant variations from this "Rule" were to be expected for certain individual numbers. Such numbers are outliers and can be either shorter or longer than the length suggested by the Rule of 8.
  • Shorter than expected Signatures:
    Consider the 25 digit number 2149199407454443676194133. Apart from its length, it doesn't appear to be anything out of the ordinary until you generate the Signature and find that it contains only two Segments:
    OEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
    and
    OEEEEEEEEEEEEEEEEEEEE.
    As you might expect this number was very carefully selected. Such numbers are not particularly common, but they do exist, and will have some impact on the value of the calculated average in any batch process in which they appear. This impact can be mitigated by increasing the number of numbers being processed in the batch.

    For a given number length, numbers which produce a signature with only a single segment are easily calculated (Four digit examples are 1365 and 5461), and signatures with other small numbers of segments are only a little harder to find.

  • Longer than expected Signatures:
    Now here is a very large number, also very carefully selected, and having 3,011 digits.

    199506311688075838488374216268358508382349683188619245485200894985294
    388302219466319199616840361945978993311294232091242715564913494137811
    175937859320963239578557300467937945267652465512660598955205500869181
    933115425086084606181046855090748660896248880904898948380092539416332
    578506215683094739025569123880652250966438744410467598716269854532228
    685381616943157756296407628368807607322285350916414761839563814589694
    638994108409605362678210646214273333940365255656495306031426802349694
    003359343166514592977732796657756061725820314079941981796073782456837
    622800373028854872519008344645814546505579296014148339216157345881392
    570953797691192778008269577356744441230620187578363255027283237892707
    103738028663930314281332414016241956716905740614196543423246388012488
    561473052074319922596117962501309928602417083408076059323201612684922
    884962558413128440615367389514871142563151110897455142033138202029316
    409575964647560104058458415660720449628670165150619206310041864222759
    086709005746064178569519114560550682512504060075198422618980592371180
    544447880729063952425483392219827074044731623767608466130337787060398
    034131971334936546227005631699374555082417809728109832913144035718775
    247685098572769379264332215993998768866608083688378380276432827751722
    736575727447841122943897338108616074232532919748131201976041782819656
    974758981645312584341359598627841301281854062834766490886905210475808
    826158239619857701224070443305830758690393196046034049731565832086721
    059133009037528234155397453943977152574552905102123109473216107534748
    257407752739863482984983407569379556466386218745694992790165721037013
    644331358172143117913982229838458473344402709641828510050729277483645
    505786345011008529878123894739286995408343461588070439591189858151457
    791771436196987281314594837832020814749821718580113890712282509058268
    174362205774759214176537156877256149045829049924610286300815355833081
    301019876758562343435389554091756234008448875261626435686488335194637
    203772932400944562469232543504006780272738377553764067268986362410374
    914109667185570507590981002467898801782719259533812824219540283027594
    084489550146766683896979968862416363133763939033734558014076367418777
    110553842257394991101864682196965816514851304942223699477147630691554
    682176828762003627772577237813653316111968112807926694818872012986436
    607685516398605346022978715575179473852463694469230878942659482170080
    511203223654962881690357391213683383935917564187338505109702716139154
    395909915981546544173363116569360311222499379699992267817323580231118
    626445752991357581750081998392362846152498810889602322443621737716180
    863570154684840586223297928538756234865564405369626220189635710288123
    615675125433383032700290976686505685571575055167275188991941297113376
    901499161813151715440077286505731895574509203301853048471138183154073
    240533190384620840364217637039115506397890007428536721962809034779745
    333204683687958685802379522186291200807428195513179481576244482985184
    615097048880272747215746881315947504097321150804981904558034168269497
    87141316063210686391511681774304792596709375

    According to the Rule-of-8 you might expect its Signature to have a little more than 24,000 segments. I won't reproduce that Signature here because it is rather large. If you really want to see it you can easily generate it using the Hailstone program.

    It has 48,126 Signature Segments which is close to twice the predicted value. So what is going on here?

    If you do generate this Signature, and if you scroll to the top of the Signature Area you will find that the first 20,000 characters consist of 10,000 Signature Segments of OE! Each of these segments will increase the subject number by a factor of 3/2. This will produce a number having many more digits than 3,011, and will therefore require many more than 24,000 segments in addition to the 10,000 already expended to get to this position.

    Needless to say, such numbers are exceedingly rare. If you were performing the Hailstone process on it manually, then after a few hundred OE segments have appeared you could be forgiven for thinking that you had discovered the holy grail of a number that would continue increasing forever. Unfortunately you would be disappointed after 10,000 OEs had appeared. Beyond that point, the Signature will revert to form and bear all the hallmarks of a totally random process.

    If occasional numbers of these types are encountered during a run of the Rule-of-8 program they will have very little effect on the overall result due to the averaging processes that occur within the program.

    Unlike shorter than expected signatures where it is easy to find a number with the shortest possible signature length, it is not so easy to find numbers with longer than expected signatures. A method of finding the number of a given length with the longest possible signature is still to be discovered.