A Solution to the Collatz Conjecture

I recently came across The Collatz Conjecture and was intrigued by how simple the puzzle was. It works like this - take an even number, divide it by two, if you end up with an odd number, multiply it by three and add one. It appears that for every number you begin with, you always ends up back at 1 through the sequence 4 - 2 - 1. Many mathematicians regard the conjecture as impossible to prove. However,  I came up with a very simple proof in half a day. 

This is a useful calculator to check various numbers and sequences :

https://www.dcode.fr/collatz-conjecture

The first step must be to begin with the simplest odd and even numbers - 1 and 2. 1 always goes to 4 and 2 always goes to 1. Every single number to infinity is comprised of these two numbers so it shouldn't really be surprising that we end up back at 4 - 2 - 1 for every number. The puzzle is set up from the start to return to this loop. 

Next, you have to understand that it is irrelevant how high the number is we start with or how high a number we reach. All that matters is the very last digit in each number. 20,036 is the same as 2,036 or 136. This is because we count to the root of 10. If you break down numbers into blocks of ten,  then 15 is simply the number 5 repeated again except it is on the second row of numbers (think of bricks in a wall). In music, there are only 12 notes, however, you can play an A note an octave higher, the musical equivalent of the second row of numbers.  Therefore, the exact same pattern or sequence will repeat itself over and over again regardless of the magnitude of the numbers. 

20036

10018

5009

15028

7514

3757

2036

1018

509

1528

764

1036

518

259

778

389

136

68

34

17

52

You will of course notice that there are variations in the sequence in those last digits. This is because different patterns repeat themselves in every second row of numbers. This flow chart gives a handy summary of the sequences :

So these variations repeat themselves and therefore the conjecture is based on very simple predictable sequences. A number ending in 6 can go to either a number ending in 8 or 3. 8 brings you to either a 9 or a 4. 9 always brings you back to an 8. Take 19 or 25,379 - you will always end up with a number ending in 8 after multiplying by 3 and adding 1. 

4 brings you to either a 7 or a 2 depending on whether the number is on every first or second row of numbers. The numbers 14, 34, 54, 74, 94, 114 etc will give you numbers ending in 7 while 24, 44, 64, 84 etc will give you numbers ending in 2. 

The next step is to check if there is a magic sequence that reduces a very high number to a very low number. This is an important step because it explains how we can go from a very high number to a low number approaching 4, 8 or 16 i.e. the final sequence in a few steps. 

 The answer is that there are two magic sequences. By continually multiplying by three and adding one, no matter what number you begin with, you will always arrive at a magic sequence eventually. Or another way of putting it, no matter how high you go, you will always return to a very low number. The first magic sequence is 8,4,2,1. This sequence arises from a simple fact that every school kid with basic knowledge of mathematics knows -  that if you divide any square of 2 in half you will always arrive at a sequence of numbers divisible by 2.  Magic sequence number 1 results in a cascade of numbers all the way down to 1. There may be other numbers in front of that 1 but this proves the tendency of the conjecture to approach 1.

The second magic sequence is 6,3,0,0,0,5 with some variation in the number of zeros. This arises out of the fact that sometimes numbers ending in 6 give a number ending in 3 which then always gives a number ending in zero. Zero numbers always half very neatly and give another cascade effect - e.g. 160, 80, 40, 20, 10 which then gives a 5. Normally, magic sequence #2 follows #1 resulting in a super cascade effect as you can see in this example. Also, if magic sequence #1 is interrupted by a 7 say, magic seq #2 will usually follow.  

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

We can see that 976 results in 61 after four steps which is 16 times less or  24   times. So magic seq#1 gives  976/24   = 61. Three steps later, we are at 46 and now locked into magic seq#2 which gives us 35 and then 160. 160 cascades to 5 after five steps, a drop of 32 times or 25  times.  Once we get to single digit 5, we will always end up back at 1. 

Here is another example. In this one, we don't reach a magic sequence for a while but once we do we are very nearly there. 

87
262
131
394
197
592
296
148
74
37
112
56
28
14
7
22
11
34
17
52
26
13
40
20
10
5
16
8
4
2
1

Some numbers like 27 just take longer to reach one of the magic sequences, 65 steps in fact. Then once the second magic sequence is reached after that, we are very close to the end. 

So there is no mystery at all here. We have endless time and limitless amounts of steps which means we will inevitably reach those magic sequences which are the simple result of counting to the base of 10 and the inherent characteristic of the number 2 and its squares. Remember that built into the rules is the number 2 - 1, 2, 4..

The only mystery is how this conjecture has puzzled mathematicians for so long.