Snowflake Numbers. A look at the Collatz Conjecture in a recreational manner. By Sir Charles W. Shults III

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1 Snowflake Numbers A look at the Collatz Conjecture in a recreational manner By Sir Charles W. Shults III For many people, mathematics is something dry and of little interest. I ran across the mention of snowflake numbers 1 in a book by Douglas R. Hofstadter (Godel, Escher, Bach, 1979) and was immediately taken by the concept. The idea is simple; a set of two rules is applied to any real number and the goal is to keep applying the rules until you reach a value of 1. What is not apparent is how amazing and complex a system emerges from this little game. It includes all real numbers and always reaches 1, although it is apparently mathematically very hard to prove that this is always true. Let me outline the rules first and then I will explain what it brings into being. Let s pick any real number and if that number is even, we will divide it by two. This will become our new number. This is rule number 1. But, if the value is odd, we will multiply it by three and add one. In this case, this will be our new number. This is rule number 2. Now, at this point we must decide if we have reached our stopping pointthe value 1. If we have not, we repeat the above process and then see if we have reached 1 yet. For example, let us pick the number 7. Since it is odd, we would apply the second rule and multiply by 3, then add 1. We end up with 22. Now, we have an even number and we apply rule number 1- divide by 2 and end up with 11. As you can see, this then leads us to 34 (through rule 2) and we then use rule 1 and get 17. This explains where the name snowflake numbers come from. Just as a snowflake will rise and fall in a cloudbank, our number values will rise and fall in an unpredictable manner- sometimes to minor steps, other times to great heights. Some people have actually preferred to call these hailstone numbers but in any event the results are the same. Now, something strange happens here, and it is this: in the end, no matter what number you may pick for the process, you will eventually end up getting to 4, 2, and 1. It does not matter what number you may start with or where it may climb or fall, because in the end every number will arrive at the bottom. But what is more amazing is that no matter the course your number may take, no matter what rising and falling it may do, it appears to be mathematically impossible to prove that every number will end at 1. This is one of those amazing and intractable logical puzzles. How do we prove our belief? How do we go about even starting? Nobody has had a clue for over seventy years 2. Using computers, nearly six quintillion values have been tried 3, and every single case has ended at the bottom. Some have undergone amazing treks into very large 1 This problem is known as the Collatz Conjecture, named after Lothar Collatz defined it in A mathematician at the University of Hamburg, Gerhard Opfer, was a student of Collatz and now feels he has solved the problem. His paper is being peer reviewed presently. 3 The actual numbers tried extend up to 5.76 x 10^18 which is close to 6 billion billion. 1 of 6

2 number territory, but in the end every one has ended at 1. Why should this be true? Again, nobody knows. The Snowflake Tree Suppose we were to try something a little different with a snowflake number. Let s look at a number and see if we can create a structure that will show how the snowflake numbers are connected and then see what emerges from this operation. The first step is to reformat the two rules. Rule 1 is our divide by 2 rule and can be written as 2/n (where n refers to the number we have chosen). Rule 2 can be referred to as 3n+1 where, again, n refers to the number we have chosen. Now we can start to unravel a little of the mystery surrounding the snowflake numbers First we can see that for any number, it may be possible to get to it from one or two different paths. What I mean is simple- some numbers can be arrived at only by rule 1 or rule 2, but some numbers can be arrived at either way. For instance, if I were to have the number 7, I know that I might have gotten 7 from applying rule 1 to the number 14. But there is no number that could have reached 7 by applying rule 2. While it is true that we can apply rule 2 to the number 2 and mathematically reach 7, it is against the rules because we must have an odd number to do that. Therefore, there is one and only one path to the number 7. What about 10? Truly we could have had 20 and applied rule 1 to get 10, but we also might have had 3 and applied rule 2. If we multiply 3 by 3 and add 1, we also get 10. So 10 can be arrived at in two ways. Here we would have a branch in our tree. Let s look at how this would appear: \ / Now, what is important is this- given a specific number, you or I cannot work backwards and determine what the previous number was in all cases. Only a portion of the cases will allow me to backtrack and see what came before. This is crucial to the whole snowflake number concept. It is not deterministic in that we cannot tell for certain the history or origin of a specific number. Conversely, without this fact we could not construct a tree of numbers. But here is the heart of the matter. It appears that every single real number, no matter what its value, will appear once and only once on this tree. So we can select any starting number and graph it as a tree, and we will find that other number tree sections will graft onto it here and there, creating the tree structure and eventually (we think) encompassing all real numbers. After a few such branches and sections have been generated, you will soon find that the bits and pieces fit together in one unique manner. Of course, depending on where you start, it may take many pieces to make out the whole, but all that is needed is to follow one of the branches to its lowest value and it will then automatically show you where it fits in the tree. 2 of 6

3 It is entertaining to graph a few random numbers and see the structure that emerges, and I have done this for a handful of values. Let s have a look and we can find out some surprising things pretty rapidly \ \ / \ \ \ / \ / \ / \ \ \ / \ \ / \ / \ / \ / \ \ / \ / \ / \ / / / / / / / (from 32) A portion of the snowflake number tree The Crucial Question So now, we can ask the most important question about snowflake numbers. Is it possible that somewhere, there is a number that does not reach 1? Is it possible that some value, instead of fitting on a tree, will produce a loop? And in a similar vein, is it 3 of 6

4 possible that somewhere there is a number (or set of numbers) that will produce another entire tree that does not intersect this one anywhere? You see, if there is a number that does produce a loop or does not reach 1, we could predict some of its properties. For instance, the number must have companions in this loop or alternate tree, because it cannot loop back on itself. How can we know this? Because if the number were odd, it could only get larger with the application of rule 2, and if the number were even, it could only get smaller with the application of rule 1. This implies that there must be at least three numbers in this alternate loop or tree. Think about the sequence at the end of the tree. Using rule 2 can only produce an even number. This is because odd times odd equals odd. When we add one more, it automatically is made even. We know that even numbers will be divided by two and so we can see that as a consequence, the shortest possible loop will be three numbers long. After all, dividing by two a couple of times produces a division by four, and multiplying by three and adding one can only equal that if you start with one- all other cases get closer and closer to only being three times as great, the larger the starting number happens to be. Another consequence of the snowflake tree is this: if any value on the existing tree emerges ever in a sequence, then it is undeniably a part of the tree. A separate snowflake tree that does not reach at its end would have to have only numbers that do not and cannot occur on the regular snowflake tree. So now we can derive another simple fact. That fact is that if a separate snowflake tree exists, it must have members that do not show up on the regular tree, and when we combine that with the reasoning about the loop at the end, we know that there must be at least four members of this tree or loop, if it exists at all. But wait- there seems to be an assumption here. That assumption is that the tree must end in a loop. That can be proven to be a fact and not an assumption. Let s see why. Suppose my tree terminated in a number. That number would be either odd or even. This dictates that we could then apply either rule 1 or rule 2 to this new number. The result will be that the new number will be either larger or smaller, and it will either already be on the tree or not. If it is not, then we were not at the end. If it is in fact on the tree, then we are in a loop. Again, the loop is an example, and it is the only known example. However, we can see that the rules will be true for any loop if any other loop is possible. Testing larger numbers with a computer soon becomes rather cumbersome because you will end up repeating steps at some point. The solution is to create a table and check off all numbers that you have previously encountered, and then you can check your results and know if you have been that way before. The path is clearly laid out for you that way, and you can stop checking a number the instant you come across it in your records. The fact is, after a few such tests by hand, it can become rather boring. The use of a computer is really the only way to go. The program for generating the numbers is simple enough, and you can literally have it step through every number automatically and fill in the table as you go. It is helpful to know that every number tested so far is on the tree, so you don t have to have a database of all numbers in your computer. You can test simply by asking if the result is smaller than, say, a trillion. If it is, there is no doubt that 4 of 6

5 the number is on the standard tree. And, numbers up into the quintillions have already been tested and all so far are on the tree. So any number that results in a value smaller than 5.76 quintillion is not left out and you will have to look further. But now we arrive at another interesting fact. If there is an alternate tree, its numbers will be in the quintillions or even larger. There is no getting around this fact. Some Facts about Snowflake Numbers It s amazing how much you can discover with a little logic and a calculator. Just looking over the last few paragraphs, it is possible to make many factual statements about snowflake numbers. For instance, for any given number, there will be an infinite number of paths to get to that number. Another interesting fact- if all numbers are on the snowflake tree, then absolutely every single number will eventually reach , but only half of the numbers will pass through 32 and only half of the numbers will pass through 5. That is the first point where the tree splits and any number you pick at random will show up only on one side or the other- no crossovers can occur. No number has yet been found that does not show up on the tree. Absolutely every number tested is on the tree somewhere, and if you were to create the tree, you would find each and every real number once and only once on it. Again, this is not provable logically at this time, and may never be. But the reasons for that are another entire article and are very interesting in their own rights. But consider that if there is a loop or branch not on the tree, then it would have a whole set of numbers associated with it and a part of it, that do not show up on the standard tree. There are two possible cases here, and one is that there is a set of numbers that are self-contained that form their own little tree or loop (and it will in fact have to end in a loop) or that the new tree will have an infinite number of members that reaches off into the unlimited supply of numbers and makes its own consistent base that is completely apart from the standard tree. The fact is, numbers are too numerous and too unknown to be able to say yet. But think about this fact. There are only 10 single-digit positive numbers, and only 90 twodigit positive numbers, and only nine hundred three-digit positive numbers. As a consequence, we can see that most numbers are more than two digits in length, since this set is only 900 members and there are infinitely many numbers. This draws us inevitably to the fact that most numbers have more than 8 million digits. That may sound odd, and you have never run across one of those numbers, but the fact is that most numbers are mind-bogglingly huge; we simply have no real-world experience with them and they are therefore never thought of. Another odd fact- if we count the steps from a starting number until it reaches 1, we will find that, in the first 100,000 numbers, one and only one will have a step number equal to its starting value, and that is 5. The number 5 takes 5 steps to reach 1 and terminate. Are there other numbers that have step counts that equal their values? Perhaps. One stumbling block to the exploration of snowflake numbers is the limitation on math that most personal computer software can handle. A number of people program in BASIC or some variety of C, others use Java and related languages. The real limit is what is known as the floating point or integer range of the software being used. BASIC, 5 of 6

6 for example, usually has a 32 bit range for its integer values. This limits the count to 2.1 billion, plus or minus. This is also true for any language that uses that 32 bit limit. This means that most people will not have the facilities to try much more than play with the lower values. However, even with no more than 110,000 or so as a starting point, the exploration will take you to over 1.5 billion as the number rises and falls. A real boon would be a language that allows you to use 64 bit notation (and even more), and some compilers now do. Personally I would recommend the Python language for new programmers as a simple to learn and use way to explore the snowflake numbers. Again, there is a limit and that is the expertise of the potential explorer of large numbers. You really need to be a programmer to reach the extremes of snowflake numbers, and few people are familiar with what it takes. So here we reach the end of this monograph, but now you have a new world to explore and play with. Have fun. 6 of 6