For Math Geeks: 3n + 1
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For Math Geeks: 3n + 1
There is one very very simple math problem that has NEVER been solved. At least no one has figured out why it works or if there are any outliers that make it not work.
The problem consists of starting with any positive (integer) number. If that number is odd, then you multiply that number by 3 and add 1. (3n +1). If that number was even, then you divide by 2.
Simple enough. Now, you take the resulting answer and do the same. If its odd use 3n +1. If its even divide by 2. Then do the same to that result. And So on.
You will ALWAYS end up in a loop of 4. Then 2. Then 1. Then back to 4. Then 2. Then 1. Etc. No matter what number you start with. There is no pattern to the numbers, they follow no logical path, but they always end up in a loop of 4, 2, 1. There are no other loops they end up in.
Since no one can prove why it works, it is assumed by most mathematicians that there must be instances where it does not work. But no one can find one.
The problem consists of starting with any positive (integer) number. If that number is odd, then you multiply that number by 3 and add 1. (3n +1). If that number was even, then you divide by 2.
Simple enough. Now, you take the resulting answer and do the same. If its odd use 3n +1. If its even divide by 2. Then do the same to that result. And So on.
You will ALWAYS end up in a loop of 4. Then 2. Then 1. Then back to 4. Then 2. Then 1. Etc. No matter what number you start with. There is no pattern to the numbers, they follow no logical path, but they always end up in a loop of 4, 2, 1. There are no other loops they end up in.
Since no one can prove why it works, it is assumed by most mathematicians that there must be instances where it does not work. But no one can find one.
- hawkfan8812
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Re: For Math Geeks: 3n + 1
Did they try using Pi?
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Re: For Math Geeks: 3n + 1
What does loop of 4 mean, my 4th result was not exactly the same as my first result, close but not equal
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Re: For Math Geeks: 3n + 1
Let's say you start with 25.hawkfan8812 wrote: ↑Wed Apr 13, 2022 3:36 pm What does loop of 4 mean, my 4th result was not exactly the same as my first result, close but not equal
Its odd, so you multiply by 3 and add 1. That would be 76.
Which is even, so you divide by 2. That would be 38.
Which is even, so you divide by 2. That would be 19.
Which is odd, so you multiply by 3 and add 1. That would be 58.
Which is even, so you divide by 2. That would be 29.
Which is odd, so you multiply by 3 and add 1. 88.
Even. Divide by 2. 44.
Even. Divide by 2. 22.
Even. Divide by 2. 11
Odd. Multiply by 3 and add 1. 34.
Even. Divide by 2. 17.
Odd. Multiply by 3 and add 1. 52.
Even. Divide by 2. 26.
Even. Divide by 2. 13.
Odd. Multiply by 3 and add 1. 40.
Even. Divide by 2. 20
Even. Divide by 2. 10
Even. Divide by 2. 5
Odd. Multiply by 3 and add 1. 16
Even. Divide by 2. 8
Even. Divide by 2. 4
Even. Divide by 2. 2
Even Divide by 2. 1
Odd. Multiply by 3 and add 1. 4
Even. Divide by 2. 2
Even. Divide by 2. 1.
Odd. Multiply by 3 and add 1. 4 ( am not in an endless loop that will go from 4 -->2 -->1 -->4 -->2 --> 1 etc.
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Re: For Math Geeks: 3n + 1
Ok, I get it now. At some point it gets to one, which then loops forever, correct?
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Re: For Math Geeks: 3n + 1
yes. But it never gets stuck in any other loop. And it never continues to increase to infinity.hawkfan8812 wrote: ↑Wed Apr 13, 2022 4:58 pm Ok, I get it now. At some point it gets to one, which then loops forever, correct?
What is odd, is if you include negative numbers, then there are several loops that the numbers can get stuck in. Start with -7. It quickly becomes stuck in a loop of -7, -20, -10, -5, -14, -7, -20 ........
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Re: For Math Geeks: 3n + 1
So in essence the formula is being changed to suit the scenario (even add 1, odd add 2).
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Re: For Math Geeks: 3n + 1
No.stonedmegman wrote: ↑Wed Apr 13, 2022 5:24 pm So in essence the formula is being changed to suit the scenario (even add 1, odd add 2).
Odd you multiply by 3 and add 1.
Even you simply divide by 2.
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Re: For Math Geeks: 3n + 1
3n•1 turns an odd number even dividing by 2 reduces the size of the number. So it not surprising the loop forms.
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Re: For Math Geeks: 3n + 1
5n • 1 works as well the loop I has more steps and the peaks are higher
Loop 1 6 3 16 8 4 2 1
Loop 1 6 3 16 8 4 2 1
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Re: For Math Geeks: 3n + 1
if you are saying to use 5n + 1 instead of 3n +1 and you somehow end up with the same pattern of always ending up in a loop of 4, 2, 1, then you would be wrong. Start with the number 27 and you end up hung in a loop with 17 being the smallest number.
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17 (over and over).
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Re: For Math Geeks: 3n + 1
No I’m saying multiplying an odd number by an odd number then adding one ends up even. Dividing by two reduces the size of the numberAnimal wrote: ↑Wed Apr 13, 2022 7:19 pmif you are saying to use 5n + 1 instead of 3n +1 and you somehow end up with the same pattern of always ending up in a loop of 4, 2, 1, then you would be wrong. Start with the number 27 and you end up hung in a loop with 17 being the smallest number.
86
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17 (over and over).
Willing to bet any odd number multiplied by a prime number plus one in place of 3n + 1 will give a loop. Loop size increases as prime numbe increases
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Re: For Math Geeks: 3n + 1
Well, let's stop right there for a second.Antknot wrote: ↑Wed Apr 13, 2022 9:47 pm
No I’m saying multiplying an odd number by an odd number then adding one ends up even. Dividing by two reduces the size of the number
Willing to bet any odd number multiplied by a prime number plus one in place of 3n + 1 will give a loop. Loop size increases as prime numbe increases
If you multiply a number by 3. Say that number is 27. and you add one to it, you obviously get an even number of 82. Then if you divide by 2 you get 41. However 41 is larger than what you started with, which was 27. In fact, if you go through the progressions of 27 you end up at 9,232 at some point. So, making a statement that you are "always" going to end up with a reduced number is simply not true. The only way you can end up with a reduced number is to end up with a series of numbers that are even (more than one).
if you don't end up with a number that divides by 2 more than once, then you are basically increasing the starting number by about 3/2.
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Re: For Math Geeks: 3n + 1
the point of 3n + 1 is that it always ends up in the exact same loop. every time. no other loop exists for this example. that's the whole point of it.Antknot wrote: ↑Wed Apr 13, 2022 9:47 pmNo I’m saying multiplying an odd number by an odd number then adding one ends up even. Dividing by two reduces the size of the numberAnimal wrote: ↑Wed Apr 13, 2022 7:19 pmif you are saying to use 5n + 1 instead of 3n +1 and you somehow end up with the same pattern of always ending up in a loop of 4, 2, 1, then you would be wrong. Start with the number 27 and you end up hung in a loop with 17 being the smallest number.
86
43
216
108
54
27
136
68
34
17 (over and over).
Willing to bet any odd number multiplied by a prime number plus one in place of 3n + 1 will give a loop. Loop size increases as prime numbe increases
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Re: For Math Geeks: 3n + 1
More than you really want to know about the Collatz conjecture.
What I find fascinating about the conjecture is the number chains it produces from some starting point. Every number produced in the list from the starting point number *also* produces the same chain list from that number's position. The other thing is that the starting point itself is also in a chain list, and so has numbers that would precede it by the function, possibly two different ones.
For example, you can start with 27, which has a chain list of 111 values until it ends in 4, 2, 1. The next number in the chain list is 82, which also has the same chain list from that value on. The number 27 can only be preceded by 54 as a new starting point, and produces the same chain list as 27 and 82. No number in the chain list can repeat, so say the number 6 will only be found once in the list, the function guarantees that.
What I find fascinating about the conjecture is the number chains it produces from some starting point. Every number produced in the list from the starting point number *also* produces the same chain list from that number's position. The other thing is that the starting point itself is also in a chain list, and so has numbers that would precede it by the function, possibly two different ones.
For example, you can start with 27, which has a chain list of 111 values until it ends in 4, 2, 1. The next number in the chain list is 82, which also has the same chain list from that value on. The number 27 can only be preceded by 54 as a new starting point, and produces the same chain list as 27 and 82. No number in the chain list can repeat, so say the number 6 will only be found once in the list, the function guarantees that.
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Re: For Math Geeks: 3n + 1
what is almost as fascinating is the guy that came up with that particular conjecture. I mean it seems pointless other than its random perfection. Unlike the perfection of the pi, which actually serves a purpose.QillerDaemon wrote: ↑Sat Apr 16, 2022 8:24 pm More than you really want to know about the Collatz conjecture.
What I find fascinating about the conjecture is the number chains it produces from some starting point. Every number produced in the list from the starting point number *also* produces the same chain list from that number's position. The other thing is that the starting point itself is also in a chain list, and so has numbers that would precede it by the function, possibly two different ones.
For example, you can start with 27, which has a chain list of 111 values until it ends in 4, 2, 1. The next number in the chain list is 82, which also has the same chain list from that value on. The number 27 can only be preceded by 54 as a new starting point, and produces the same chain list as 27 and 82. No number in the chain list can repeat, so say the number 6 will only be found once in the list, the function guarantees that.