Hello again,

This is more of a phenomenon than a trick though, but quite interesting. Makes you wonder if there is something special that somehow got encoded in the universe. We depend highly on math in science so when something unusual comes up it kind of takes on a meaning of it's own.

I'll start by listing an illustration of the phenomenon.
We start with the powers of 2:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

Next, for each number above we add the digits so for example 32 becomes 5. However, if we end up with more than one digit in the result, we do the same with that, so to get the total it may have to be done several times for the same number.
For example, 128 => 11 => 2 final result.
Working from left to right i get:
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, ...

Notice the pattern 1, 2, 4, 8, 7, 5 that keeps repeating and this goes on forever.

What else is interesting is that there is no 3, 6, or 9. The inventor N. Tesla called the 3-6-9 numbers special.

What we did above was add the digits of each number over and over until we got a single digit result.
If we do the same thing with powers of 3:
1, 3, 9, 27, 81, 243, etc.

we get:
1, 3, 9, 9, 9, 9, 9, 9, 9, 9, forever.
And to top that off, you can always tell if a larger number is divisible by 9 by applying this logic.
For example, is 243 whole number divisible by 9? Lets see (we did this above though):
2+4+3=9
and since 9 is divisible by 9 that means 243 is divisible by 9.
How about a big number, 81243?
8+1+2+4+3+7=25
so that is not divisible by 9.
How about the number 111105?
That one is easy, 1+1+1+1+5=9
so that number is divisible by 9.
How about a much bigger number, 1853020188851841?
Well this takes two steps:
1+8+5+3+2+1+8+8+8+5+1+8+4+1=63
and then
6+3=9
so that big number is also divisible by 9.

So it seems there is something else at work, but who knows what. I just call it the magic of integers, for now.

There are also other similar tricks that makes you think there is something deeper embedded in math itself
Maybe it is just that there is so much structure in the universe that it shows up now and then in the math.
I think we can say that structure implies symmetry and symmetry is something we usually notice and consider to be special.

 

Hello again,

This is more of a phenomenon than a trick though, but quite interesting. Makes you wonder if there is something special that somehow got encoded in the universe. We depend highly on math in science so when something unusual comes up it kind of takes on a meaning of it's own.

I'll start by listing an illustration of the phenomenon.
We start with the powers of 2:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

Next, for each number above we add the digits so for example 32 becomes 5. However, if we end up with more than one digit in the result, we do the same with that, so to get the total it may have to be done several times for the same number.
For example, 128 => 11 => 2 final result.
Working from left to right i get:
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, ...

Notice the pattern 1, 2, 4, 8, 7, 5 that keeps repeating and this goes on forever.

What else is interesting is that there is no 3, 6, or 9. The inventor N. Tesla called the 3-6-9 numbers special.

What we did above was add the digits of each number over and over until we got a single digit result.
If we do the same thing with powers of 3:
1, 3, 9, 27, 81, 243, etc.

we get:
1, 3, 9, 9, 9, 9, 9, 9, 9, 9, forever.
And to top that off, you can always tell if a larger number is divisible by 9 by applying this logic.
For example, is 243 whole number divisible by 9? Lets see (we did this above though):
2+4+3=9
and since 9 is divisible by 9 that means 243 is divisible by 9.
How about a big number, 81243?
8+1+2+4+3+7=25
so that is not divisible by 9.
How about the number 111105?
That one is easy, 1+1+1+1+5=9
so that number is divisible by 9.
How about a much bigger number, 1853020188851841?
Well this takes two steps:
1+8+5+3+2+1+8+8+8+5+1+8+4+1=63
and then
6+3=9
so that big number is also divisible by 9.

So it seems there is something else at work, but who knows what. I just call it the magic of integers, for now.

There are also other similar tricks that makes you think there is something deeper embedded in math itself
Maybe it is just that there is so much structure in the universe that it shows up now and then in the math.
I think we can say that structure implies symmetry and symmetry is something we usually notice and consider to be special.

Does it also work in base 9 or base 11?

 

There is nothing particularly special about 3, 6, and 9, other than the fact that all of those numbers divide 9, which is simply one less than the number base used to represent decimal numbers. Choosing base 16 (ie. hexadecimal) instead would make 5, 10, and 15 "special". And of course, selecting a number base that is one greater than a prime number would yield a situation where only the prime itself would be "special".

Anyway, the reason behind the divisibility "trick" is essentially due to the above relationship between any given number base and its value minus one. The decimal number 81243 for example can obviously be broken down as (8 * 10^4 + 1 * 10^3 + 2 * 10^2 + 4 * 10^1 + 3 * 10^0). This in turn implies a mathematical structure which essentially "embeds" the remainder of that particular number divided by nine, right there within the decimal representation itself. But again, the principle can be easily extended to any number base whatsoever.

 

Does it also work in base 9 or base 11?

I havent tried that at all. Have not tried anything else yet.

 

There is nothing particularly special about 3, 6, and 9, other than the fact that all of those numbers divide 9, which is simply one less than the number base used to represent decimal numbers. Choosing base 16 (ie. hexadecimal) instead would make 5, 10, and 15 "special". And of course, selecting a number base that is one greater than a prime number would yield a situation where only the prime itself would be "special".

Anyway, the reason behind the divisibility "trick" is essentially due to the above relationship between any given number base and its value minus one. The decimal number 81243 for example can obviously be broken down as (8 * 10^4 + 1 * 10^3 + 2 * 10^2 + 4 * 10^1 + 3 * 10^0). This in turn implies a mathematical structure which essentially "embeds" the remainder of that particular number divided by nine, right there within the decimal representation itself. But again, the principle can be easily extended to any number base whatsoever.

Well Tesla thought 3, 6 and 9 where special, not sure why though. It is probably not just because of that. I think he wanted certain things in his life to be divisible by 3 also. Dont know why.

 

Well Tesla thought 3, 6 and 9 where special, not sure why though. It is probably not just because of that. I think he wanted certain things in his life to be divisible by 3 also. Dont know why.

Draw a circle marked from 1 to 9 around it's circumference. Now using powers of two as you did in your first example, draw lines from one point to the next taken from the list of remainders.

Spoiler: And the result?

That's right, we get Tesla's familiar "vortex" diagram!

 

Draw a circle marked from 1 to 9 around it's circumference. Now using powers of two as you did in your first example, draw lines from one point to the next taken from the list of remainders.

Spoiler: And the result?

View attachment 262777


That's right, we get Tesla's familiar "vortex" diagram!

Is this going to be on the test? (chuckle)

Yes that's it! Thanks a bunch. You get a double "W" or something on the graphic when you draw that.