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Awani
09-22-2009, 08:08 PM
I dabbled a bit in this before and I found it interesting that the following pattern always emerge...

If you take any number, lets say 418 and then add up all its combinations like:

418
814
148
184
481
841

Total = 2886 = 2+8+8+6 = 6

Now you do the same with 417 = 9, 416 = 3, 415 = 6, 414 = 9 etc...

The pattern is always 3, 6, 9... magic numbers.

Nothing new, but interesting!

:cool:

Zephyr
09-22-2009, 09:13 PM
Most curious. At what point does this pattern begin? As soon as the 10 basic digits are exhausted?
11+11=22-->4
12+21=33-->6
13+31=44-->8
14+41=55-->10-->1
...
25+52=77-->14-->5
...
56+65=121-->4
...
98+89=187-->16-->7
99+99=198-->18-->9
.
100+010+001=111-->3
101+101+110+110+011+011=444-->12-->3
102+201+120+210+021+012=666-->18-->9
...

Well, it looks like the crunched sums of single digits go up in increments of one, in two digit, they go up in increments of two, and in three, they would go up in increments of 3. Howevever we see that "101" and "102" are a counterexample to this pattern. I wonder why?

*Zephyr*

LeoRetilus
12-13-2009, 05:38 AM
Interesting, sometimes I find myself with a calculator in my hand and find simliar patterns, except I always find myself multipying 9 times other numbers and resolving the sum back to nine.

For example: 9x9x6x7=3402 -> 3+4+0+2=9
9x4x5x3x9x5=24300 ->2+4+3+0+0=9
6x9x21x15x3=51030 ->5+1+0+3+0=9
and on and on , but it does stop at one point then begin again

The magical number 9:)

Ghislain
12-13-2009, 08:31 AM
All multiples of nine resolve back to nine.

It is the principle whereby you predict what
a person is going to say.

Try this on someone:

tell them to pick a number
now multiply the number by nine
if the resulting number has more than one digit
keep adding the digits together until it resolves
to one digit (which you know will be nine)
get them to subtract 5 from their total
(they now have four)
tell them to associate the number with a letter
from the alphabet (a=1,b=2.....) they now have
"d"
tell them to choose a country (in english) starting
with their letter. It is usually Denmark
tell them to choose an animal starting with the
second letter of the country name. (this is
usually an elephant.
ask then to think of what colour they associate with
this animal (this is usually grey)
ask them if they are thinking of a grey elephant
from denmark...watch the surprise on their face
when you get it right :)

When they have got to the country you could ask them to choose
the last letter, "k", and think of an animal. This is usually a
kangaroo or a koala, now ask them to think of a fruit begining with
the last letter of the animal then tell them it is either an apple or an orange.

Ghislain

vigilance
02-15-2019, 11:13 PM
The internet always references Tesla and his quote on 3, 6, 9 (if he really said it, many attributions popular on the internet are spurious):

“If you only knew the magnificence of the three, the six and the nine… then you would have a key to the universe.”

But it's also mentioned in Aesch Mezareph (the author is talking about the columns of his squares):

'And all the columns and lines, as well from the bottom to the top, as from the right to the left, and from one angle to another, give the same sum and thou mayest vary the same ad infinitum. And the various totals always observe this principle, that their lesser number is always 3, 9, or 6 ; and again, 3, 9 or 6 and so on. Concerning which I could reveal many things to thee.'

I found this out, when dealing with the factors of 360 - whole number divisions of the circle. The digital root of the angles when dividing by 3, 6, 12, 15, 24, 30, and 60 always come out to 3, 6 or 9. The basic pattern is 3, then bisected for 6, then bisected for 12, and then 24. But also 15, 3x5, and those bisected for 30 and again for 60.

http://i.imgur.com/i4HJH5S.jpg

It doesn't work out for 9 or 18. 9 Does it's own thing, with 9-18-36 eventually running a pattern of 1 to 9. 45 runs the pattern 9-to-1 instead.

http://i.imgur.com/l18u4Ow.jpg

And factors of 4 and 5 that don't figure in elsewhere gives 9s.

http://i.imgur.com/yFyeDQ3.jpg

Tannur
10-12-2020, 10:09 PM
Interesting pattern Awani.

Notice how the sum of digits required is just 222(a+b+c) for any generic three digit number "abc". This is because when we permute the digits 6 times as shown by Awani, we end up with the a digit appearing exactly twice in each position (eg for a in the 100th position, it appears once for abc and once for acb - exactly twice in 100th position). So overall, we end up with 200a, 20a, 2a and similarly for b,c; ie overall 222(a+b+c).

Now notice also that the sum of digits of a number when applied over and over again is the same as its remainder when divided by 9. For example, for 4768 the remainder is 529*9+7 - remainder 7. And indeed, if we add digits over and over again we get 4+7+6+8=25 and 2+5=7. It isn't too difficult to prove this.

So for 222(a+b+c) we see that the sum of digits is just remainder when divided by 9. For the largest three digit number, this is 222(9+9+9), ie it is a multiple of 9, so it can only be 9. So it starts off at 9. Then notice that when we subtract 1 to get 998, the overall sum of the 6 permutations will just be 222(9+9+8). In other words, in terms of the remainder when divided by 9, we are just losing 222, or a remainder of 6 (since this is the digit sum of 222). So from 9, we go down to 9-6=3. So for 998 the answer is 3.

So really we are just working with remainders when divided by 9. 9, 3, then 3-6=-3, but as a remainder with respect to 9 this is 9-3=6. So 9,3,6. We can carry on like this to get 9,3,6,9,3,6,...

So this also applies to 4 digit numbers but the digit sum this time is the remainder of 6666(a+b+c+d) when divided by 9. It starts off with 9999 giving 6666(9+9+9+9) which leads to 9 (since divisible by 9). Then 6666(9+9+9+8) we lose 6666 which is just 24 or 2+4=6 with respect to 9. So as before, we start with 9 and lose 6 each time. So it is exactly the same pattern as before: 9,6,3,9,... You can verify this by actually carrying out the additions.

vigilance
10-12-2020, 10:13 PM
Take a number.. 72 subtract the opposite 27..

72
27

45

It'll always be a factor of 9. There's a name for this I believe. don't know what it is.. i don't think its "the rule of 9"

Tannur
10-12-2020, 10:29 PM
Take a number.. 72 subtract the opposite 27..

72
27

45

It'll always be a factor of 9. There's a name for this I believe. don't know what it is.. i don't think its "the rule of 9"
Nice. This is similar to Awani's one.

I don't know the rule but if we consider a random number "ab" it is really 10a+b. E.g. 72=10*7+2. Its permutation is 10b+a. Subtracting we get 9(a-b), which is a multiple of 9. But above I stated the rule that the remainder when a number is divided by 9 is the same as its ultimate digit sum when divided by 9 (eg 47567 - remainder when divided by 9 is 2 and its digit sum is 29 and 2+9=11 and 1+1=2). Hence digit sum of subtraction result will always be a multiple of 9.

vigilance
10-12-2020, 11:12 PM
Nice. This is similar to Awani's one.

I don't know the rule but if we consider a random number "ab" it is really 10a+b. E.g. 72=10*7+2. Its permutation is 10b+a. Subtracting we get 9(a-b), which is a multiple of 9. But above I stated the rule that the remainder when a number is divided by 9 is the same as its ultimate digit sum when divided by 9 (eg 47567 - remainder when divided by 9 is 2 and its digit sum is 29 and 2+9=11 and 1+1=2). Hence digit sum of subtraction result will always be a multiple of 9.

Are you familiar with Ars Combinatoria? Art of Memory? Abulafia in terms of mystical permutating? etc? Lull's system?

Tannur
10-13-2020, 08:13 AM
Are you familiar with Ars Combinatoria? Art of Memory? Abulafia in terms of mystical permutating? etc? Lull's system?
Yes, yes, no, no.

In my opinion, magic and things like that mean nothing without gnosis.

vigilance
10-13-2020, 04:30 PM
Yes, yes, no, no.

In my opinion, magic and things like that mean nothing without gnosis.

Let me know what you think.

Tannur
10-19-2020, 02:35 PM

Let me know what you think.
Nice video. Quite biased towards the "autistic" side of maths though. I don't deny that many mathematicians are autistic but there's a huge difference between mathematicians and maths itself.

vigilance
10-19-2020, 03:12 PM
yeah.. that's the left side. The "master". strictness and all that. It needs the right side, too. I was so like that. I honestly only skimmed your math. THATS what it looked like to me.

I was mentioning Bruno and Lull.. its left and right. I'd like to see more of what you do. I'd really follow it this time.

Tannur
10-19-2020, 04:56 PM
yeah.. that's the left side. The "master". strictness and all that. It needs the right side, too. I was so like that. I honestly only skimmed your math. THATS what it looked like to me.

I was mentioning Bruno and Lull.. its left and right. I'd like to see more of what you do. I'd really follow it this time.
Why do you feel that Bruno and Lull's works are even mathematics to begin with? No serious mathematician would consider numerology, number symbolism or sacred geometry to be mathematics. They probably wouldn't deny their consistency and truth as separate disciples in themselves, but definitely not mathematics. On the other hand, no scientist would be able to deny Alchemy if it was openly revealed to him, because it is entirely scientific, only deepening its profundity and significance.

This is what I was attempting to show through the example of my work in Number Theory. I was working entirely within high school mathematics, and yet the result I found had immense spiritual significance, at least for me. As I said in that thread, I believe that this was precisely because of how I approached mathematics: without knowing it, I was attempting to reach the "ultimate" root of a particular problem - the one source of that problem from which the problem was completely explained and revealed. Problem solving doesn't work like this. It works by playing around with a problem until you discover a "crack" somewhere. Then you explore this further, always with the aim of solving the problem in question, but not exploring it further to reach its "root". Such ideas are completely foreign to the world of olympiad mathematics I was working within. I like how Grothendiek explains this in the following passage:

"Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle -- while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone."

I couldn't explain it better myself. I completely resonate with what he says about people being faster and picking up ideas easily. I definitely was not one of those people and really struggled to even get to the stage where I could solve olympiad problems. But I stopped being interested in learning more and more theorems very early on and just kept exploring guided by my own spiritual longing. This is what eventually led to my work with that problem I talk about in my other post.

Why I am mentioning all of this? As I said in that thread I posted about my Number Theory research, I was just trying to give a blueprint or "philosophy" through which Alchemy can be understood. Alchemy is like that problem I encountered, but a more "absolute" version of it. Whilst that problem does not reach the absolute root of mathematics, Alchemy *does* reach the absolute root of the physical world - of the four elements. And it is in its focus on this root that Alchemy acquires its spiritual flavour and significance. Without this focus, it collapses back into ordinary Aristotelian natural philosophy. Similarly, if I hadn't attempted to reach the root of that problem (thus forging my own "relative" alchemical path within Number Theory), my solution would have collapsed into mere competition level mathematics. That may have led to a quick, "elegant" solution to the problem like others on the forum in which I encountered it, but at the cost of the more profound work I was to accomplish through it.

The dream visions I had during my work on that problem, the sheer euphoria, the all-pervasive feeling of being at the doors of a huge castle burning with desire to pass onto the other side, the constant influx of illuminations and eureka moments in waking consciousness (they tend to always occur subconsciously when one is *away* from the problem), and so on. It was definitely right brain mathematics alright, but mathematics it was - not numerology or some other science. Similarly, alchemy is science alright, just like modern science. The only difference is that it seeks the root, the origin, the source. That is what happened to me back in high school without me even realizing what was happening at the time. That is what happened to Grothendiek. That is what happened to the alchemists.

Note: I did not intend to write an essay. Seems I am more enthusiastic about this topic than I thought!

vigilance
10-20-2020, 01:24 AM
This is what I found. I stopped when I realized it was already know. I'm going to have to upgrade my computer, and start getting into C#. For some reason I think it's going to be easy.

Thursday, May 16, 2013 at 11:13am EDT
Fractions with prime denominators 

A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The period of the repeating decimal of 1/p is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the period is equal to p − 1; if not, the period is a factor of p − 1. This result can be deduced from Fermat's little theorem, which states that 10p−1 = 1 (mod p).

The base-10 repetend (the repeating decimal part) of the reciprocal of any prime number greater than 5 is divisible by 9.[3]

i dont have a lot of time for an involved reply right now, but I'd like to revisit these topics with you at a later date. Gmail, facebook, here. Whereever is most comfrotable for you.