CONTENTS
 Introduction to the Collatz Conjecture. 
 Introducing Signatures and Syllables. 
 A tutorial on Signatures and Syllables. 
 Signature / Syllable Analysis. 
 The Collatz Rule of 8. 
 Appendix 1 - Novel behaviors in the Collatz series. 
 Appendix 2 - Signature / Number conversion Algorithm. 
 Appendix 3 - An Algorithm for deriving a long Signature. 
 Appendix 4 - Getting your copy of the Collatz / Crossword Express program. 

• If the current number is even, divide by 2 to generate the next member.
• If the current number is odd, multiply by 3 and add 1 to generate the next member.

• The Collatz series for 1:
  

  1  4  2  1

• The Collatz series for 3:
  

  3  10  5 16  8  4  2  1

• The Collatz series for 7:

  7  22  11  34  17  52  26  13  40  20  10  5  16  8  4  2  1

• The Collatz Series for 27:

    27    82    41   124    62    31    94    47   142   71    214   107  322   161
   484   242   121   364   182    91   274   137   412   206   103   310  155   466
   233   700   350   175   526   263   790   395  1186   593  1780   890  445  1336
   668   334   167   502   251   754   377  1132   566   283   850   425 1276   638
   319   958   479  1438   719  2158  1079  3238  1619  4858  2429  7288 3644  1822
   911  2734  1367  4102  2051  6154  3077  9232  4616  2308  1154   577 1732   866
   433  1300   650   325   976   488   244   122    61   184    92    46   23    70
    35   106    53   160    80    40    20    10     5    16     8     4    2     1

OEOEEOEOEOEOEOEEOEEOEOEEOEOEOEEOEOEOEOEEOEEEOEOEOEEOEOEE OEOEOEOEOEOEEEOEOEOEOEEEEOEEOEEOEEEEOEEEOEOEOEEEEEOEEEEO

A further change in the presentation is to insert a space immediately before each O in the series. This breaks the series up into fragments called Syllables. The number of Syllables in a Signature is a very important factor in the analysis of the Collatz process as you will see in subsequent sections of this article.

OE OEE OE OE OE OE OEE OEE OE OEE OE OE OEE OE OE OE OEE OEEE OE OE OEE OE OEE OE OE OE OE OE OEEE OE OE OE OEEEE OEE OEE OEEEE OEEE OE OE OEEEEE OEEEEO

It should be noted that all Signatures in this article begin with O, which implies that only odd numbers are of interest. This is because application of the Collatz process to an even number immediately reveals an underlying (and smaller) odd number.

• Final Signature.
  If you apply the Collatz process repetitively to a number, adding the letters E or O as appropriate to the Signature string as you go until you encounter the digit 1, then the Signature string which results will be a Final Signature.

• Partial Signature.
  A Partial Signature results when the Collatz process is interrupted before a digit 1 has been encountered. Alternatively, you can create a Partial Signature using a text editor by typing a string consisting only of Os and Es. The first and last letters must both be O, and there must not be any examples of consecutive Os. It is most unlikely that a Signature created in this way will be of the Final variety.

• The Signature of a series is a string of letters which correspond to the numbers of the series, with an O for each odd number and an E for each even number.

• The members of most Collatz series go up and down in what appears to be a totally random fashion in much the same way as a Hailstone rises and falls in a storm cell until it grows to a such a size and weight that it has no alternative but to fall to the ground. Because of this, the numbers considered here are often referred to as Hailstone Numbers

• Most commentaries on the Collatz conjecture warn of the unpredictable and haphazard behavior of the number series it produces. This is undoubtedly a fact, but by the time you have finished studying this article, you will see that some of the behaviors are reassuringly regular and very predictable and that the Normal or Gaussian Distribution of statistics is a remarkably accurate predictor for some of this behavior.

• The first of the above series appears to be unique. It is the only one which returns to its starting value.

• The number 10 appears in both the second and third examples. This will happen to any number which is half of some even number, whilst also being one more than three times some odd number.

• Theoretically, there are two ways in which a number could fail to return to 1, and thereby fail the conjecture. It could continue its up and down behavior forever, or it could enter a loop, and circulate around that loop forever. In either event, an infinitely long Signature would ensue, and this would quickly become apparent to the Rule-of-8 process (see later) if ever it should happen to encounter such a number.

• Signatures can vary greatly in length. The longest Signature for any number less than 10¹⁷ has 2091 letters. But after you have studied the Tutorial on Signatures and Syllables you will be able to design and build numbers which have vastly longer signatures than this. There is no upper limit to the length of a Signature.

• The algorithm for calculating a Signature from a Number is defined quite simply by the two rules stated at the beginning of this page. The algorithm for performing the reverse operation is considerably more intricate, and although an understanding of its operation is not essential, a complete listing of it is provided in  Appendix 2. 

• It will be very much to your advantage if you have a working copy of the Collatz software on your computer as you study this section of the article, so before you proceed any further, please refer to  Appendix 4.  and complete the Download and Installation instructions. Try to have the program showing on the screen at the same time as you read the contents of this tutorial.

• Calculate a Signature from a Number. Type the number directly into the Number field, and click the Number to Signature button. The Signature will appear in the Signature field, and some explanatory notes will appear in the Collatz Results field.

• Calculate a Number from a Signature. Type the signature directly into the Signature field, and click the Signature to Number button. The Number will appear in the Number field, and the Collatz Results field will be populated with an abbreviated description of the algorithm used to calculate the number. Note especially the format of the number display. It represents an infinite series of numbers, all of which have signatures which begin with a common set of characters.

• How to design a number which has very special characteristics. For example say you wanted a number which would start its Collatz series with ten consecutive odd numbers, followed by ten consecutive even numbers, followed by another ten odd numbers. All you need to do is to create a signature with ten consecutive OE syllables followed by a syllable containing ten consecutive Es, followed again by ten consecutive OE syllables. In other words, the complete signature would be
  OEOEOEOEOEOEOEOEOEOE
  EEEEEEEEEE
  OEOEOEOEOEOEOEOEOEOE
  Type it into the Signature field, and click the Signature to Number button. You should receive the number 451306495. Use a calculator to extract the Collatz series for this number, and you will find that the series begins with the odd and even numbers appearing in exactly the order you specified.

• Here is a number for you to ponder. It is quite ordinary except for the fact that it has 3,011 digits. An interesting experiment for you to try, is to copy the array of digits into the clipboard of your computer and then paste them into the Number window of the Collatz program. Clicking the Number to Signature button will start the calculation of the Signature. Depending on your computer it may take quite a bit of time (think in terms of minutes rather than seconds), but the interesting result will be well worth the wait. You will find that the first 20,000 characters of the signature will consist of 10,000 OE Syllables!

  19950631168807583848837421626835850838234968318861924548520089498529438830221946631919961
  68403619459789933112942320912427155649134941378111759378593209632395785573004679379452676
  52465512660598955205500869181933115425086084606181046855090748660896248880904898948380092
  53941633257850621568309473902556912388065225096643874441046759871626985453222868538161694
  31577562964076283688076073222853509164147618395638145896946389941084096053626782106462142
  73333940365255656495306031426802349694003359343166514592977732796657756061725820314079941
  98179607378245683762280037302885487251900834464581454650557929601414833921615734588139257
  09537976911927780082695773567444412306201875783632550272832378927071037380286639303142813
  32414016241956716905740614196543423246388012488561473052074319922596117962501309928602417
  08340807605932320161268492288496255841312844061536738951487114256315111089745514203313820
  20293164095759646475601040584584156607204496286701651506192063100418642227590867090057460
  64178569519114560550682512504060075198422618980592371180544447880729063952425483392219827
  07404473162376760846613033778706039803413197133493654622700563169937455508241780972810983
  29131440357187752476850985727693792643322159939987688666080836883783802764328277517227365
  75727447841122943897338108616074232532919748131201976041782819656974758981645312584341359
  59862784130128185406283476649088690521047580882615823961985770122407044330583075869039319
  60460340497315658320867210591330090375282341553974539439771525745529051021231094732161075
  34748257407752739863482984983407569379556466386218745694992790165721037013644331358172143
  11791398222983845847334440270964182851005072927748364550578634501100852987812389473928699
  54083434615880704395911898581514577917714361969872813145948378320208147498217185801138907
  12282509058268174362205774759214176537156877256149045829049924610286300815355833081301019
  87675856234343538955409175623400844887526162643568648833519463720377293240094456246923254
  35040067802727383775537640672689863624103749141096671855705075909810024678988017827192595
  33812824219540283027594084489550146766683896979968862416363133763939033734558014076367418
  77711055384225739499110186468219696581651485130494222369947714763069155468217682876200362
  77725772378136533161119681128079266948188720129864366076855163986053460229787155751794738
  52463694469230878942659482170080511203223654962881690357391213683383935917564187338505109
  70271613915439590991598154654417336311656936031122249937969999226781732358023111862644575
  29913575817500819983923628461524988108896023224436217377161808635701546848405862232979285
  38756234865564405369626220189635710288123615675125433383032700290976686505685571575055167
  27518899194129711337690149916181315171544007728650573189557450920330185304847113818315407
  32405331903846208403642176370391155063978900074285367219628090347797453332046836879586858
  02379522186291200807428195513179481576244482985184615097048880272747215746881315947504097
  32115080498190455803416826949787141316063210686391511681774304792596709375

  This means that if you were to perform the Collatz process on this number, then each time you multiplied by 3, added 1 and divided by 2 you would get another odd number, and this would continue for an amazing 10,000 times. Thereafter, the pattern would cease, and the normal mixture of odd and even numbers would return, until the number 1 is finally encountered as it always is.

  Some additional results will appear in the Collatz Results window as shown below:-
  The initial number has 3011 digits.
  The number of Signature letters is 134405
  The number of Signature Syllables is 48126
  The biggest number encountered in the Collatz series is 3262700 - - - - 4400000
  This Number contains 4772 digits

• Try some signatures of your own choosing. You should be able to create numbers with some very unlikely characteristics. Relax about the size of the numbers involved. The program should have no problem whatever handling numbers with at least a few thousand digits. There is no end to the interesting games you can play in this way.

• In case you are wondering... Well I hope you are wondering about how one finds a number of over 3000 digits which begins with a string of 10,000 OE syllables. Clearly, this is not the result of any sort of a search routine. It is done by crafting a signature which consists of 10,000 OEs and using the Signature / Number conversion program to obtain the desired number.

• Towards an infinitely long signature It has already been mentioned that an exception to the Collatz Conjecture demands the existence of an infinitely long Signature, so you may be thinking that as long as you can isolate a Number which produces a Signature designed to your specification, you might be able to isolate one which generates an infinitely long signature. The facts however are that although you can create signatures of any given length, infinitely long signatures will apparently always remain beyond your grasp.

Every Syllable begins with OE, which implies:-

• Multiplication by 3.
• Addition of 1.
• Division by 2.

Following the division by 2, our number may be odd or even with equal probability.

This implies that:-
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.

And so on...

Clearly, the probability decreases by a factor of 2 for each E added to the Syllable, so longer Syllables are progressively less likely. However they do have a greater impact on the Collatz process due to the greater number of divisions by 2.

The reasoning presented above is captured in tabular form in the following:-.

• Signature Syllables.
  A list of possible Signature Syllables which contain varying numbers of Es from 1 up to 16. Syllables longer than this will of course be encountered when very large numbers are submitted to the Collatz process. Numbers having thousands of digits will be met and processed in later sections of this article.

• [D]ivisions by 2.
  This is simply the number of Es contained within the Syllable.

• [P]robability.
  The probability of a random Signature Syllable being of this type. As discussed previously, each Syllable type has a probability of half that of the previous Syllable type. The sum of the probabilities in this column will approach a value of 1 as the list is extended.

• [D]x[P].
  The product of the number of divisions by 2 and the probability of this Syllable being generated. The resulting number provides a measure of the overall likelihood of achieving a division by 2 by means of this Syllable when a Syllable is generated.

• [D]x[P] (normalized)
  The numbers in this column have exactly the same values as the numbers in the previous column, but they have been normalized so that each number has a denominator of 65536. When we see a number such as 6144 / 65536, it tells us that when 65536 Syllables of a Signature are generated, 6144 of the divisions by 2 will be generated by Syllables which have the form OEEEEEE.

To see some experimental evidence which supports this rule, refer to  Result 2: Profile of the Signature Syllable data of 100 digit numbers. 

A very interesting circumstance arises when we calculate a series of 8 consecutive Signature Syllables. 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 = .10011

In short it gives us a division by a number very close to 10. This in turn translates to a reduction of one in the number of digits in the number being processed. To a first approximation then, an n digit number will generate n*8 Signature Syllables on its journey to the expected concluding value of 1. This is the basis for what I call The Rule of 8 which will be demonstrated quite soon. When you study this topic, I believe you will be pleasantly surprised at how closely numbers right across the vast number spectrum obey this rule.

• The 3 in the mathematical statement is always accompanied by the addition of 1. This is not expected to cause a big departure in the operation of the Rule, but it is always present, and the departure is always in the same direction.

• The 4 is the average calculated in the Signature Syllable analysis discussed previously. On any given Syllable calculation it will in fact be some power of 2, but averaged over a large number of calculations the probabilities involved will dictate that the outcome will be a division by very close to 4.

• The mathematical statement above doesn't give us exactly one tenth, although it is, fortuitously, remarkably close to that figure. As a result, we are entitled to be quietly confident that the Rule of 8 will be closely observed.

On the page dealing with  Signature / Syllable Analysis.  mention was made of the fact that as the Collatz process operates on a number, it will reduce the length of that number by roughly one digit for each group of 8 Signature Syllables which it calculates for that number. For a single number, this is of course just an approximation, but the Rule-of-8 demonstration discussed here allows you to execute the Collatz process on a large group of numbers all of the same length. When the results are averaged over a group of some (or many) thousands of numbers, a quite remarkable result emerges as you will soon see.

The demonstration is part of the Rule-of-8 program, and is best described by reference to the typical output shown in the following graphic.

As the demonstration runs using the defaults supplied, the program applies the collatz process to one million 30 digit numbers, and in so doing produces a set of one million signatures for those numbers. The important information here is not the content of the signature, but its length. The program maintains a list of signature lengths, and as the signature length of each number is determined, the list item for that length is incremented by one. When all one million numbers have been processed this list can be used to display the histogram you see in the graphic. Naturally, as the program runs, it provides an indication of progress by redrawing the histogram at regular intervals. This can be quite spectacular to watch, and is recommended for your entertainment.

It is hoped that the following dot points will add meaning to what you see in the above graphic:-

• The X axis of this histogram is calibrated in terms of signature lengths which range between 64 and 654. These figures are also included in tabular form above the histogram.

• The Y axis is calibrated between 0 and 9937, indicating that the longest signature achieved by any of the one million 30 digit number was in fact achieved by a total 9937 of them.

• A downward pointing red arrow below the x axis points to the location of the average value of Signature Syllables predicted by the Rule of 8, and an upward pointing blue arrow labeled Average points to the location of the current average for the numbers processed so far. It is quite entertaining to watch this indicator as the program runs. Initially it wanders up and down but quickly settles down to a number very close to 240 as predicted by the Rule of 8.

• Although 64 is the shortest signature length encountered in this particular demonstration, it would be quite wrong to assume that 64 is the shortest possible signature for 30 digit numbers. In fact, there is always at least one, and often two numbers which will result in a signature having only one syllable. In the case of 30 digit numbers, the numbers 422550200076076467165567735125 and 105637550019019116791391933781 will collapse to 1 with a single Syllable Signature. These numbers look impressively large, but considerably less so when expressed as (2¹⁰⁰ - 1) / 3 and (2⁹⁸ - 1) / 3.

• Similarly, although 654 was the longest signature encountered in the demonstration, it is almost certainly not the longest possible signature. Unlike the shortest signature, finding the longest signature seems not to be a trivial matter. This could be fertile ground for people who can't resist a mathematical challenge.

• The shape of this histogram will strike a chord with anyone who has more than a passing interest in the subjects of probability and statistics. The graph is immediately recognizable as a bell curve, or "normal" distribution with some obvious differences. The main difference is that it is very far from being the smooth curve normally expected. It seems that certain Signature Syllable count values are favored by the Collatz process while others are disadvantaged. Why this is so may be another interesting question for additional research in the future.

• Another point worth noting is the way in which the right extreme of the curve seems to extend out much further than the left extreme. For the time being we will just assume this is because there is an absolute lower limit of 1 for the numbers which can appear in the left portion this histogram, while the right portion might possibly extend indefinitely, as would be the case if an exception to the Collatz Conjecture were to be encountered.

• The Normal Distribution curve is again firmly in evidence with the great majority of data points in this histogram crowded very compactly around the value of 720, which is the value suggested by the Rule of 8.

• The curve of the histogram thins out very substantially as it approaches both the left and the right extremes of the graphic.

• The thinning of the curve mentioned above continues to the left, to the extent that the last of the results shrink to only a single pixel.

• The shortest recorded Signature has a Syllable length of 364. This is most certainly not the shortest possible signature. It is mentioned elsewhere in this article that, for numbers having a given number of digits (90 in this case), there will always be at least one example of a signature length of just one syllable. The likelihood of encountering such a number in a run of the Rule of 8 program on 90 digit numbers is as good as zero. It is the same as the likelihood of selecting one particular atom out of 10⁹⁰ atoms or, to put it another way, one particular atom out of all the atoms in 10 billion copies of the observable universe. I venture to suggest that this is not very likely.

• The stretched signatures are drawn using red, and clearly show the continued reduction in frequency.

• The Y axis of the histogram is calibrated in terms of frequency. The most frequently encountered signature length was 719, and was encountered 5598 times.

• The thinning of the curve also continues to the right, with the last Signature having a Syllable length of 1295. This is very likely not the longest signature possible for numbers having 90 digits. Finding longer signatures than this is not such a simple matter as finding the shortest possible signature. More research is required to settle this question.

• In the Introduction to the Collatz problem it was mentioned that any number which fails the Collatz conjecture will be characterized by an infinite length signature. The rapid thinning out of longer signatures here doesn't bode well for the prospects of finding an infinitely long one which is required to disprove the Collatz conjecture.

• In order to obtain a more detailed view of the extreme right hand end of the Normal Distribution Curve for Numbers having exactly 100 digits, the program was modified to provide an output of the Signature Length, and the Number which produced that Signature for any cases in which the Signature Length was greater than 1400.

  The following paragraph contains a list of forty 100 digit numbers, all of which yield signatures having lengths of more than 1400 syllables. It may appear from this that they are rather plentiful, and we shouldn't be surprised about this since they are selected randomly from a collection of 10¹⁰⁰ 100 digit numbers. Although they are quite plentiful, they are fiendishly hard to find. You can find them using the Rule of 8 function of the Collatz program if you use the option function of that program to request a run of say 10 million 100 digit numbers, and request that any number which produces a signature of over 1400 syllables be displayed in the Collatz Report section of the screen. Typically, a run of 10 million numbers will produce two or three examples of signature lengths in excess of 1400. On the other hand it is not unusual to find no such examples at all.

  Patience is a virtue.

• The first 20 signatures are associated with signature lengths which range between 1402 and 1420, a span of only 19 lengths.

• The range of lengths occupied by the second group of 20 signatures extends from 1420 up to 1504, a span of 84. This demonstrate clearly that the density of signatures is decreasing quite rapidly.

• In the first group of signatures there are six examples of signature lengths which play host to multiple signatures, while in the second group of 20 there are only two such examples.

• The highest signature length discovered was 1504. There are very likely longer signature lengths than this, but searching for them would be a very time consuming and probably unrewarding task.

• It would be foolish to insist that there are no examples of 100 digit numbers with signature lengths greater than 1504, but it is surely unlikely that there exists a 100 (or less) digit number with an infinitely long signature.

• Which brings us to the crucial point, (mentioned in several other places in this report), that an exception to the Collatz Conjecture demands the existence of a finite number which has an infinitely long signature.

• The above results apply to numbers having "only" 100 digits. I have used the Rule of 8 program to perform the same process on 1000 digit numbers, and also to a lesser extent on 10,000 digit numbers, and the result is the same in all cases.

The progressive results provided by the Rule of 8 function include a profile of the entire set of Signature Syllables 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 Syllables should be 8 x 100 x 1,000,000 = 800,000,000. The figure that is reported by the program is 799,108,217. 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 Syllable types. Note that almost exactly half of all the Syllables are of type OE, and that all of the subsequent types progressively decrease by a factor of close to 2. This justifies the equal probability assumptions made in the topic  Signature / Syllable Analysis. 

     0     521864777009781972171399170839     1565594331029345916514197512518         [1]
     0     521864777009781972171399170839     2348391496544018874771296268778         [1]
     0     521864777009781972171399170839     3522587244816028312156944403168         [1]
     1     323420776575166478500093050537     3683964783176505669165122403790         [1]
     1     323420776575166478500093050537     5525947174764758503747683605686         [1]
     1     323420776575166478500093050537     8288920762147137755621525408530         [1]
     1     323420776575166478500093050537     12433381143220706633432288112796        [2]
     1     323420776575166478500093050537     13987553786123294962611324126898        [2]
     1     323420776575166478500093050537     20981330679184942443916986190348        [2]
     1     323420776575166478500093050537     23603997014083060249406609464144        [2]
    57     916855648438295922460724587307     24472957408552401667321569784324        [2]
    87     621623765988457188136100529945     28709147380980211756617196305502        [2]
    87     621623765988457188136100529945     43063721071470317634925794458254        [2]
    87     621623765988457188136100529945     64595581607205476452388691687382        [2]
    87     621623765988457188136100529945     96893372410808214678583037531074        [2]
    87     621623765988457188136100529945     145340058616212322017874556296612       [3]
    87     621623765988457188136100529945     183946011686143720053872485312906       [3]
    87     621623765988457188136100529945     275919017529215580080808727969360       [3]
   357     126481844367082466405697580063     384123720446478525524194192690150       [3]
   357     126481844367082466405697580063     576185580669717788286291289035226       [3]
   357     126481844367082466405697580063     864278371004576682429436933552840       [3]
   997     930817431324349698729704998367     1242071656860735704986768744595224      [4]
  3759     833314741525573657655193087439     1687177372739226871344116090500816      [4]
  4790     802879304996710367566694233327     2002334859286487864932913145951352      [4]
  4790     802879304996710367566694233327     2534205056284461204055718200344688      [4]
 12498     724912411722761965880444096681     3478991944441262688411679665259144      [4]
 13198     288842135708910560684485601487     4995984046651086558655411738847260      [4]
 13198     288842135708910560684485601487     5620482052482472378487338206203170      [4]
 13198     288842135708910560684485601487     8430723078723708567731007309304756      [4]
 18895     337595270660394052978947869065     9122282670150866104209365939507014      [4]
 18895     337595270660394052978947869065     13683424005226299156314048909260522     [5]
 18895     337595270660394052978947869065     20525136007839448734471073363890784     [5]
 46838     877683494233739625091193455399     23082073760820664932033630627369574     [5]
 46838     877683494233739625091193455399     34623110641230997398050445941054362     [5]
 46838     877683494233739625091193455399     51934665961846496097075668911581544     [5]
 56900     808023609957127685989776095391     58102268660421757049325755468362498     [5]
 56900     808023609957127685989776095391     87153402990632635573988633202543748     [5]
121196     305282207143663775670173201499     123597536981034324928986406162487428    [6]
121196     305282207143663775670173201499     160725630931863703306657628708270410    [6]
121196     305282207143663775670173201499     241088446397795554959986443062405616    [6]
121196     305282207143663775670173201499     257451382945302186864008960438223202    [6]
121196     305282207143663775670173201499     386177074417953280296013440657334804    [6]
159448     620544853285226825440776099267     557672052424846300330533042985763488    [6]

• Column 1: The number of numbers which had been processed before processing was begun on this one.

• Column 2: The thirty digit number currently being processed.

• Column 3: The value of the current number in the Collatz series.

• Column 4: The number of additional digits in the Collatz series number, compared to the number being processed.

    When n=0, the initial number is 23.
    O   E   O   E    O   E    E   E   E   E   O   E   E   E   E   O
    23  70  35  106  53  160  80  40  20  10  5   16  8   4   2   1

    When n=1, the initial number is 4096 + 23 = 4119.
    O    E     O    E     O    E     E     E    E    E    O    E
    4119 12358 6179 18538 9269 27808 13904 6952 3476 1738 869  2608
    E    E     E    O
    1304 652   326  163

It has been mentioned several times in this report that it is an easy matter to derive large numbers which have quite short signatures. However the inverse operation of finding the longest possible signature for numbers which have a (relatively) small number of digits is far from simple. This appendix will describe a method of obtaining relatively long examples.

Consider the following diagram of a number map. Virtually everyone who knows anything about the Collatz conjecture could start at 1442 and calculate their way down to the 1. It is simply a matter of applying the Collatz process to generate the familiar sequence of numbers, but how do you get back to 1442 if you start from, for example, 5. The difficulty is that at each number you have a choice of two operations. You can always multiply by 2, but if the current number is 1 modulo 3, you can also subtract 1 and divide by 3. How do you decide which choice to make?

The aim of the process is to get the longest possible signature, and to achieve this you must take the divide by 3 option as often as possible.

^^^^

1442

^^^^                          721

1624 - 541 - 1082 - 2164 >

812

406

^^^             203

916 - 305 - 610 >

458

^^^                                    229

130 - 43 - 86 - 172 - 344 - 688 >

^^^                    65

148 - 49 - 98 - 196 >

74

^^                         37

22 - 7 - 14 - 28 - 112 >

^^           11

52 - 17 - 34  >

26

^^                       13

16 - 5 - 10 - 20 - 40 >

4

2

1

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Appendix 4.
Getting your copy of the Collatz / Crossword Express program

The Collatz program which supports the principles discussed in this article was generated over a period of some years while the principles themselves were emerging from my research. During this time, a program called Crossword Express was also being developed, and at the time it seemed appropriate to incorporate the much shorter Collatz program into the main body of Crossword Express. As a result, if you go the extra mile to download and install this program you will receive a significant bonus of an extensive puzzle generation platform which I believe surpasses the performance of all competing products, and it won't cost you a cent.

The entire package is written in the Java programming language, and so is able to be installed and run on both Apple and Windows computers. There are some differences in the steps involved between these two systems, and so what follows are two separate paragraphs which provide all of the details you will need to achieve a satisfactory result.

Downloading, Installing and Configuring Crossword Express for Apple computers.

• Click this link to download the file CrosswordExpress.zip into the standard location which your browser uses for downloaded files.

•  Drag the newly downloaded zip file onto the Desktop.

• Unzip the file by means of a double click on the icon of the .zip file. This will give you a folder called CrosswordExpress. You can now safely delete the CrosswordExpress.zip file.

• Move the CrosswordExpress folder to the location on your hard drive where you would like it to be stored permanently. The Applications folder would be a good choice, but the decision is up to you.

• Double click the CrosswordExpress folder in the new location you chose for it, right click the Crossword-Express.jar file, and select Open With followed by Jar Launcher.app. This is a once only operation which in effect tells the computer that the application being opened is a trusted application.

• Double click the CrosswordExpress folder in the new location you chose for it, then right click the Crossword-Express.jar file, and select Make Alias. This will give you a file called Crossword-Express.jar alias.

• Drag the Crossword-Express.jar alias file to an easily accessible location, so that you can start Crossword Express and make use of the Collatz program whenever the need arises. The Desktop would be a good choice, but once again, the choice is yours.

---

Downloading, Installing and Configuring Crossword Express for Windows computers.

• Click this link to download the file CrosswordExpress.zip into the standard location which your browser uses for downloaded files.

•  Drag the newly downloaded zip file onto the Desktop.

• This is where the recommended procedure for Apple and Windows diverge. Thirty years of experience with Crossword Express has revealed that there are many unzip utilities available for use with Windows, and many of these have a faithful band of regular users. Consequently, I make no firm recommendation on the subject of unzipping this file, except to say that when you are finished, the Desktop must contain a new folder called CrosswordExpress. You can now safely delete the CrosswordExpress.zip file.

• Move the CrosswordExpress folder to the location on your hard drive where you would like it to be stored permanently. The Programs folder would be a good choice, but the decision is up to you.

• Double click the CrosswordExpress folder in the new location you chose for it, right click the Crossword-Express.jar file, and select Open With followed by Java (TM) Platform SE binary. This is a once only operation which in effect tells the computer that the application being opened is a trusted application.

• Double click the CrosswordExpress folder in the new location you chose for it, then right click the Crossword-Express.jar file, and select Create shortcut. This will give you a file called Crossword-Express.jar-Shortcut.

• Drag the Crossword-Express.jar-Shortcut file to an easily accessible location, so that you can start Crossword Express and make use of the Collatz program whenever the need arises. The choice is yours.

---

Possible Installation Difficulties.

The executable file of Crossword Express is CrosswordExpress.jar Such files (having a .jar extension) will run on any computer which has a Java Runtime Environment (JRE) installed. If it fails to run on your computer, it simply means that your computer has not yet been equipped with a JRE. This is easily remedied:-

• For Apple
  Select Settings / Java / Update The resulting dialog will tell you the version of Java you have installed. If this is not the recommended version, an Update button will appear, and you can use this to download the correct version. For detailed assistance on the subject, visit https://java.com/en/download/help/mac_install.xml

• For Windows.
  Select Start / Control Panel / Java / Update. Clicking the Update Now button will either tell you that you have the correct version already installed, or it will automatically install the correct version. For detailed assistance on the subject, visit https://java.com/en/download/faq/java_win64bit.xml

---

And here is the payoff for your care and attention.

To start the Collatz program, select Collatz from the list of puzzle options, and then click and GO. This will present you with a window similar to this:-

The window that you see will be larger than this, but all of the components of the window are resizeable using simple mouse operations.

When you are studying Number / Signature Conversions, you will be working with the Number window, the Signature window, and the associated green arrow buttons.

When you are studying The Rule-of-8 you will be using the Rule-of-8 Demonstration menu option.