Factorial pattern

 Posts: 5420
 Joined: Sun Aug 31, 2014 7:39 am
Factorial pattern
1! = 1
3! = 2•3
5! = 4•5•6
7! = 7•8•9•10
The dot means times.
This pattern doesn't extend beyond 7! How this pattern could exist (plus others) has no current explanation because whoever came up with the notation, the algebra and other ideas couldn't have planned for this pattern (btw I checked the internet and there's no mention about this pattern).
The table above easily leads to other equations. For example, we have:
1!•3!•5! = 6!
1!•3!•5!•7! = 10! = 6!•7!
PhilX
3! = 2•3
5! = 4•5•6
7! = 7•8•9•10
The dot means times.
This pattern doesn't extend beyond 7! How this pattern could exist (plus others) has no current explanation because whoever came up with the notation, the algebra and other ideas couldn't have planned for this pattern (btw I checked the internet and there's no mention about this pattern).
The table above easily leads to other equations. For example, we have:
1!•3!•5! = 6!
1!•3!•5!•7! = 10! = 6!•7!
PhilX
Re: Factorial pattern
What pattern is that? Your expression for 3! starts with 2; your expression for 5! starts with 4; but your expression for 7! starts with 7, not 6. And your expression for 1! starts with 1, not 0. There's no pattern at all, even in the tiny data set you gave.Philosophy Explorer wrote:1! = 1
3! = 2•3
5! = 4•5•6
7! = 7•8•9•10
The dot means times.
This pattern doesn't extend beyond 7!
But you know numbers are full of coincidences. Take a circle. A line joining two points of the circle is called a chord.
If you draw no chords, the interior of the circle consists of 1 region.
If you draw 1 chord, it divides the interior of the circle into 2 regions.
If you draw 2 chords, you get 4 regions.
If you draw 3 chords, you get 8 regions.
If you draw 4 chords, you get 16 regions.
How many regions do you think you get if you draw 5 chords?
That's right. 31.
There's even a name for this phenomenon: The strong law of small numbers. It says that there aren't enough numbers to go around, so coincidences pop up all the time. You might enjoy this terrific little paper that lists all kinds of patterns that hold up to very high values, until they fail.
http://www.maa.org/sites/default/files/ ... 97712.pdf
Last edited by wtf on Wed Apr 05, 2017 7:31 pm, edited 1 time in total.

 Posts: 5420
 Joined: Sun Aug 31, 2014 7:39 am
Re: Factorial pattern
The left side has the odd factorials which is increasing from 1! to 7! The right side starts with 1, the next line increases to 2 x 3, the third line increases by 1 again (in the leadoff number) to 4 which is multiplied by the next two natural numbers 5 and 6, and the last line sees its leadoff number increased by 1, 7, which is multiplied by the next three natural numbers, 8, 9 and 10 plus the numbers on the righthand side form a triangle. And you can't see this pattern???wtf wrote:What pattern is that? Your expression for 3! starts with 2; your expression for 5! starts with 4; but your expression for 7! starts with 7, not 6. And your expression for 1! starts with 1, not 0. There's no pattern at all, even in the tiny data set you gave.Philosophy Explorer wrote:1! = 1
3! = 2•3
5! = 4•5•6
7! = 7•8•9•10
The dot means times.
This pattern doesn't extend beyond 7!
To mention, when you said "tinydata set", that implies you know what I'm talking about and you agree with me there is a pattern.
PhilX
Re: Factorial pattern
How do you figure that? I don't know what you're talking about, I denied there's a pattern, I pointed out that math is full of numerical coincidences, and I linked an entertaining and readable paper that talks about numerical coincidences.Philosophy Explorer wrote: To mention, when you said "tinydata set", that implies you know what I'm talking about and you agree with me there is a pattern.
Yes the right side is cute but the "pattern" last for four items. That's a small data set. You're right, there's a "pattern" in the same sense that the chords of a circle form the pattern 1, 2, 4, 8, 16, till the next number breaks the pattern. That's not a pattern, that's a coincidence.

 Posts: 5420
 Joined: Sun Aug 31, 2014 7:39 am
Re: Factorial pattern
Now you're contradicting yourself. You're saying the first five numbers make a pattern (which I agree with). Next you're saying the 6th number breaks the pattern and is just a coincidence. But the pattern must exist to be broken, otherwise it can't be broken. So in your example, you do have a pattern after all (as I do in mine).wtf wrote:How do you figure that? I don't know what you're talking about, I denied there's a pattern, I pointed out that math is full of numerical coincidences, and I linked an entertaining and readable paper that talks about numerical coincidences.Philosophy Explorer wrote: To mention, when you said "tinydata set", that implies you know what I'm talking about and you agree with me there is a pattern.
Yes the right side is cute but the "pattern" last for four items. That's a small data set. You're right, there's a "pattern" in the same sense that the chords of a circle form the pattern 1, 2, 4, 8, 16, till the next number breaks the pattern. That's not a pattern, that's a coincidence.
PhilX
Re: Factorial pattern
Yes you're entirely right. There's a little pattern till there isn't. Just like with the chords. You are correct that I did not initially see your point. Now that I do, the chord example and the paper I linked are still relevant.Philosophy Explorer wrote: Now you're contradicting yourself. You're saying the first five numbers make a pattern (which I agree with). Next you're saying the 6th number breaks the pattern and is just a coincidence. But the pattern must exist to be broken, otherwise it can't be broken. So in your example, you do have a pattern after all (as I do in mine).

 Posts: 5420
 Joined: Sun Aug 31, 2014 7:39 am
Re: Factorial pattern
I was already aware of the chord example you brought up.wtf wrote:Yes you're entirely right. There's a little pattern till there isn't. Just like with the chords. You are correct that I did not initially see your point. Now that I do, the chord example and the paper I linked are still relevant.Philosophy Explorer wrote: Now you're contradicting yourself. You're saying the first five numbers make a pattern (which I agree with). Next you're saying the 6th number breaks the pattern and is just a coincidence. But the pattern must exist to be broken, otherwise it can't be broken. So in your example, you do have a pattern after all (as I do in mine).
Some patterns are finite and others are infinite. I consider patterns as guidelines for further exploration which can lead to exciting math. Which branches of math appeal to you?
PhilX
Re: Factorial pattern
Unpatterns. Mathematical objects that have no patterns at all. The noncomputable numbers, for example. Without them you couldn't do standard math, yet there is absolutely no pattern to them.Philosophy Explorer wrote:
I was already aware of the chord example you brought up.
Some patterns are finite and others are infinite. I consider patterns as guidelines for further exploration which can lead to exciting math. Which branches of math appeal to you?
In general I'm drawn to set theory, elementary real analysis, some abstract algebra. These days elementary computability theory and its relationship to the philosophy of the real numbers. Never had a high interest in number theory or numeric patterns.

 Posts: 5420
 Joined: Sun Aug 31, 2014 7:39 am
Re: Factorial pattern
Besides number theory, I'm drawn to calculus, differential equations, linear algebra and some topology plus some algebra too. I like to experiment and explore. Sometimes I turn up something which is worth further exploration (at least to me). When people say they don't like math or find it hard, sometimes I wonder what they base their assessment on?wtf wrote:Unpatterns. Mathematical objects that have no patterns at all. The noncomputable numbers, for example. Without them you couldn't do standard math, yet there is absolutely no pattern to them.Philosophy Explorer wrote:
I was already aware of the chord example you brought up.
Some patterns are finite and others are infinite. I consider patterns as guidelines for further exploration which can lead to exciting math. Which branches of math appeal to you?
In general I'm drawn to set theory, elementary real analysis, some abstract algebra. These days elementary computability theory and its relationship to the philosophy of the real numbers. Never had a high interest in number theory or numeric patterns.
PhilX
Re: Factorial pattern
For some reason my eyes glazed over in differential equations. It's funny how different parts of math appeal to different people. I found out years later that my DiffEQ professor was very famous. If I'd been paying attention I might have learned something.Philosophy Explorer wrote: Besides number theory, I'm drawn to calculus, differential equations, linear algebra and some topology plus some algebra too. I like to experiment and explore. Sometimes I turn up something which is worth further exploration (at least to me). When people say they don't like math or find it hard, sometimes I wonder what they base their assessment on?
It's also interesting that I'm not experimental at all. I'm more drawn to understanding the formalisms. These days when I commit to doing some math, I always want to learn more about particular area that's just beyond my level. Some more abstract algebra or trying to understand a particular proof or theorem. Reading accessible math papers I find online.
I don't actually spend any time just playing around with numbers or geometric figures, and I don't think I ever did. I fell in love with math when I had a great teacher for Euclidean geometry in high school. I don't think I had any insight or even much interest in sideangleside or computing similar triangles. I was drawn to the logical structure of the proofs.
Re: Factorial pattern
In HS I did very well in Geometry because I wrote everything down and it helped me memorize everything, I did very well in that class. The next year I took trigonometry and there was one girl in the class taking advanced math, the teacher would give her separate assignments to do during class. The period was split by lunch and we would have half a class, go to lunch, and come back for the rest of the class. One day I came back and the girl who was taking advanced math was working on a problem, I looked over her shoulder and then told her the answer, she just looked at me and said "You're pretty smart". Later everyone took the SAT's, (It was a small school), and I got the highest total and math score, the girl who was taking advanced math got a slightly higher verbal score.wtf wrote: I fell in love with math when I had a great teacher for Euclidean geometry in high school. I don't think I had any insight or even much interest in sideangleside or computing similar triangles. I was drawn to the logical structure of the proofs.
BTW I was accepted at one State Teachers College as a math major, but I decided to go for Industrial Arts, I have often wondered How things might have been different. I might have flunked out as a math major.

 Posts: 5420
 Joined: Sun Aug 31, 2014 7:39 am
Re: Factorial pattern
It sounds unusual to me to have a mixed math class covering trigonometry and advanced math. Was there a reason behind that?thedoc wrote:In HS I did very well in Geometry because I wrote everything down and it helped me memorize everything, I did very well in that class. The next year I took trigonometry and there was one girl in the class taking advanced math, the teacher would give her separate assignments to do during class. The period was split by lunch and we would have half a class, go to lunch, and come back for the rest of the class. One day I came back and the girl who was taking advanced math was working on a problem, I looked over her shoulder and then told her the answer, she just looked at me and said "You're pretty smart". Later everyone took the SAT's, (It was a small school), and I got the highest total and math score, the girl who was taking advanced math got a slightly higher verbal score.wtf wrote: I fell in love with math when I had a great teacher for Euclidean geometry in high school. I don't think I had any insight or even much interest in sideangleside or computing similar triangles. I was drawn to the logical structure of the proofs.
BTW I was accepted at one State Teachers College as a math major, but I decided to go for Industrial Arts, I have often wondered How things might have been different. I might have flunked out as a math major.
PhilX
Re: Factorial pattern
Small school and only one student to take advanced math.Philosophy Explorer wrote: It sounds unusual to me to have a mixed math class covering trigonometry and advanced math. Was there a reason behind that?
PhilX
Re: Factorial pattern
My daughter graduated from R.I.T. magna cum laude, and I graduated from Millersville state teachers college "thankyou laude", literally I had to talk my way out, the staff wasn't going to allow me to graduate, till I pointed out that I had fulfilled the requirements of the old system.
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