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Aug 24, 1999, 3:00:00 AM8/24/99

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Hello all.

Do you know of anyway of generating sequences of random numbers in your head

(without becoming predictable - ie no repeating patterns)? Especially

sequences of 3 possible numbers (0,1,2 perhaps) so that each term has a 1/3

probability of occurring.

No dice,coins,watches or other physical tools - although observing another

person's action may be acceptable and have potential.

One could do things like

Ri+1 = (ARi + B) mod N

using the like of A=317537 B=4378591 and N=3

but I'm looking for something that needs only moderate arithmetic computing

capability. ie in the head

And A & B smallish don't give very good results.

Warren

Aug 24, 1999, 3:00:00 AM8/24/99

to

If pi is normal, then of course you just choose a random digit of that, and it

can be 10 possible values: 1,2,3,4,5,6,7,8,9,0. If it is 1,4,7, or 0, choose

random possibility #1. if 2,5, or 8, possibility #2. if 3,6, or 9, choose

possibility #3. if 0, pick again. but all this is assuming pi is normal...

Sam "Main Night" Alexander

Cast your vote for Freedom! ! !

(you must be 18 and a U.S. citizen to vote)

can be 10 possible values: 1,2,3,4,5,6,7,8,9,0. If it is 1,4,7, or 0, choose

random possibility #1. if 2,5, or 8, possibility #2. if 3,6, or 9, choose

possibility #3. if 0, pick again. but all this is assuming pi is normal...

Sam "Main Night" Alexander

Cast your vote for Freedom! ! !

(you must be 18 and a U.S. citizen to vote)

Aug 24, 1999, 3:00:00 AM8/24/99

to

I'm not sure, but I think actual counts of the digits of pi up to 1 000 000

digits have shown that pi seems to be normal, although we have no proof of

it ... it is a very interesting question.

digits have shown that pi seems to be normal, although we have no proof of

it ... it is a very interesting question.

Is there any theorem which shows a relation between a number's normalcy to

base x and base y? I.e. : if a number is normal, using base 10, will it be

normal using any other base?

Ernest

Main Night <main...@aol.com> wrote in message

news:19990824061109...@ng-fx1.aol.com...

Aug 24, 1999, 3:00:00 AM8/24/99

to

In article <19990824061109...@ng-fx1.aol.com>, main...@aol.com

says...

>

>If pi is normal, then of course you just choose a random digit of that, and it

>can be 10 possible values: 1,2,3,4,5,6,7,8,9,0. If it is 1,4,7, or 0, choose

>random possibility #1. if 2,5, or 8, possibility #2. if 3,6, or 9, choose

>possibility #3. if 0, pick again. but all this is assuming pi is normal...

>Sam "Main Night" Alexander

>Cast your vote for Freedom! ! !

>(you must be 18 and a U.S. citizen to vote)

says...

>

>If pi is normal, then of course you just choose a random digit of that, and it

>can be 10 possible values: 1,2,3,4,5,6,7,8,9,0. If it is 1,4,7, or 0, choose

>random possibility #1. if 2,5, or 8, possibility #2. if 3,6, or 9, choose

>possibility #3. if 0, pick again. but all this is assuming pi is normal...

>Sam "Main Night" Alexander

>Cast your vote for Freedom! ! !

>(you must be 18 and a U.S. citizen to vote)

Thanks for your thoughts.

But what I'm (essentially) trying to do, is find an "easy" way of finding a

sequence of "reasonably" random numbers when I don't have access to a computer,

books, or any other tools.

And remembering enough digits of PI is beyond my modest memory - and how do I

choose a random digit randomly!?

I suspect that the digits of pi would be close enough to draws from a uniform

distribution which is what's required.

Warren

Aug 24, 1999, 3:00:00 AM8/24/99

to

Main Night wrote:

> If pi is normal, then of course you just choose a random digit of that,

That's silly. If you have the ability to choose a "random digit of pi"

you could just use the position of the digit you chose for your random

number - introducing pi adds nothing. Answers to "how can I choose a

random [whatever]" should probably not begin "choose a random..."

Aug 24, 1999, 3:00:00 AM8/24/99

to

Warren Shallcross wrote:

&&&&&&&&&&&&&&&&&&&&&&&&&&

Choose a 2-digit number, say 23, your "seed".

Form a new 2-digit number:

the 10's digit plus 6 times the units digit.

The example sequence is

23 --> 20 --> 02 --> 12 --> 13 --> 19 --> 55 --> 35 --> ...

and its period is the order of the multiplier, 6, in the group of

residues relatively prime to the modulus, 10. (59 in this case).

The "random digits" are the units digits of the 2-digit numbers,

ie, 3,0,2,2,3,9,5,... the sequence mod 10.

The arithmetic is simple enough to carry out in your head.

This is an example of my "multiply-with-carry"

random number generator, and it seems to provide quite satisfactory

sequences mod 2^32 or 2^64 , particularly well suited to

the way that modern CPU's do integer arithmetic.

You may choose various multipliers and moduli for examples of random

selection of the types you ask about.

A description of the multiply-with-carry method is in the postscript

file mwc1.ps, included in

The Marsaglia Random Number CDROM

with

The DIEHARD Battery of Tests of Randomness,

available at

http://stat.fsu.edu/pub/diehard/

George Marsaglia

Aug 24, 1999, 3:00:00 AM8/24/99

to

On 24 Aug 1999 08:52:35 GMT, fo...@uq.net.au (Warren Shallcross) wrote:

>Hello all.

>

>Do you know of anyway of generating sequences of random numbers in your head

>(without becoming predictable - ie no repeating patterns)? Especially

>sequences of 3 possible numbers (0,1,2 perhaps) so that each term has a 1/3

>probability of occurring.

In your head is a tough requirement. The only math I could do in my

head is to use a mnemonic as with pi: "see I have the rhyme

assisting, etc" counting the number of letters per word. Perhaps you

could do a word association sequence and count the letters as you go.

The tough part would be to match Benford's law.

See:

http://www.newscientist.com/ns/19990710/thepowerof.html

an article in New Scientist which discusses how the statistics of

usual numbers can be used to catch people who try to cook books.

Given that, another consideration might be to mix it up a bit, while

watching out for standard word length frequencies. Possibly,

something like count up for the first word, then count back down for

the second, etc.

John

Aug 24, 1999, 3:00:00 AM8/24/99

to

Hi, what does normal mean?

Thanks,

Dan Pilloff

Main Night <main...@aol.com> wrote in message

news:19990824061109...@ng-fx1.aol.com...

> If pi is normal, then of course you just choose a random digit of that,

Aug 25, 1999, 3:00:00 AM8/25/99

to

Warren Shallcross <fo...@uq.net.au> wrote:

> One could do things like

>

> Ri+1 = (ARi + B) mod N

>

> using the like of A=317537 B=4378591 and N=3

> One could do things like

>

> Ri+1 = (ARi + B) mod N

>

> using the like of A=317537 B=4378591 and N=3

Just in passing, this is not particularly unpredictable. You would not

want to use such a linear congruential generator for anything where you're

worried about someone else predicting the output.

Aug 25, 1999, 3:00:00 AM8/25/99

to

In article <7pu24h$nem$1...@news.adamastor.ac.za>, "Ernest Lötter"

<1277...@narga.sun.ac.za> wrote:

<1277...@narga.sun.ac.za> wrote:

> Is there any theorem which shows a relation between a number's normalcy to

> base x and base y? I.e. : if a number is normal, using base 10, will it be

> normal using any other base?

There are theorems, due to Wolfgang Schmidt, Andy Pollington, and others,

which say, roughly speaking, that given any two sets of bases, there are

numbers that are normal to all the bases in the first set and to none of

the bases in the other. The "roughly speaking" is to cover obvious

counterexamples when, e.g., one base is a power of another.

Gerry Myerson (ge...@mpce.mq.edu.au)

Aug 25, 1999, 3:00:00 AM8/25/99

to

In article <7ptmgj$jue$2...@bunyip.cc.uq.edu.au>, fo...@uq.net.au (Warren

Shallcross) wrote:

Shallcross) wrote:

> Hello all.

>

> Do you know of anyway of generating sequences of random numbers in your head

> (without becoming predictable - ie no repeating patterns)? Especially

> sequences of 3 possible numbers (0,1,2 perhaps) so that each term has a 1/3

> probability of occurring.

How about using the base 3 analogue of Champernowne's number? Write the

numbers 1, 2, 3, ... in base 3, and concatenate:

121011122021221001011021101111121201211222002012022102112122202212221000...

For variety, instead of starting at 1, start at 100 (base 10, = 1201,

base 3):

120112021212121112121220122112222000200120022010201120122020....

Gerry Myerson (ge...@mpce.mq.edu.au)

Aug 25, 1999, 3:00:00 AM8/25/99

to

In article <37c2fbb0...@news.frontiernet.net>, jmb...@frontiernet.net

says...

>

>On 24 Aug 1999 08:52:35 GMT, fo...@uq.net.au (Warren Shallcross) wrote:

>

>>Hello all.

>>

>>Do you know of anyway of generating sequences of random numbers in your head

>>(without becoming predictable - ie no repeating patterns)? Especially

>>sequences of 3 possible numbers (0,1,2 perhaps) so that each term has a 1/3

>>probability of occurring.

says...

>

>On 24 Aug 1999 08:52:35 GMT, fo...@uq.net.au (Warren Shallcross) wrote:

>

>>Hello all.

>>

>>Do you know of anyway of generating sequences of random numbers in your head

>>(without becoming predictable - ie no repeating patterns)? Especially

>>sequences of 3 possible numbers (0,1,2 perhaps) so that each term has a 1/3

>>probability of occurring.

>In your head is a tough requirement. The only math I could do in my

>head is to use a mnemonic as with pi: "see I have the rhyme

>assisting, etc" counting the number of letters per word. Perhaps you

>could do a word association sequence and count the letters as you go.

>The tough part would be to match Benford's law.

>See:

>http://www.newscientist.com/ns/19990710/thepowerof.html

>an article in New Scientist which discusses how the statistics of

>usual numbers can be used to catch people who try to cook books.

>Given that, another consideration might be to mix it up a bit, while

>watching out for standard word length frequencies. Possibly,

>something like count up for the first word, then count back down for

>the second, etc.

>

>John

>head is to use a mnemonic as with pi: "see I have the rhyme

>assisting, etc" counting the number of letters per word. Perhaps you

>could do a word association sequence and count the letters as you go.

>The tough part would be to match Benford's law.

>See:

>http://www.newscientist.com/ns/19990710/thepowerof.html

>an article in New Scientist which discusses how the statistics of

>usual numbers can be used to catch people who try to cook books.

>Given that, another consideration might be to mix it up a bit, while

>watching out for standard word length frequencies. Possibly,

>something like count up for the first word, then count back down for

>the second, etc.

>

>John

John

Thanks for that.

I like the words idea; with a poem, perhaps, should be able to find a traversal

method for getting a reasonable sequence of numbers.

Regards

Warren

Aug 25, 1999, 3:00:00 AM8/25/99

to

In article <19990824061109...@ng-fx1.aol.com>, main...@aol.com (Main Night) writes:

> If pi is normal, then of course you just choose a random digit of that, and it

> can be 10 possible values: 1,2,3,4,5,6,7,8,9,0. If it is 1,4,7, or 0, choose

> random possibility #1. if 2,5, or 8, possibility #2. if 3,6, or 9, choose

> possibility #3. if 0, pick again. but all this is assuming pi is normal...

> If pi is normal, then of course you just choose a random digit of that, and it

> can be 10 possible values: 1,2,3,4,5,6,7,8,9,0. If it is 1,4,7, or 0, choose

> random possibility #1. if 2,5, or 8, possibility #2. if 3,6, or 9, choose

> possibility #3. if 0, pick again. but all this is assuming pi is normal...

Of course, this scheme does not work. There is no uniform distribution

across a countably infinite set.

All we know is that if pi is normal then as the leading substring of pi

from which we randomly and uniformly select is extended without

bound, this scheme becomes arbitrarily good. But we have no guarantee

that the scheme is any good at all for any particular number of leading

digits of pi.

John Briggs vax...@alpha.tst.tracor.com

Aug 25, 1999, 3:00:00 AM8/25/99

to

In article <7pv9hi$515$1...@moonbeam.aecom.yu.edu>,

A number (in particular an irrational number, since rational numbers never have

this property) is normal in base b iff in the base b expansion of the number,

every finite digit sequence appears at least one. As stated earlier in this

thread, the number 0.012345678901020304... is trivially normal base 10, but

not very useful. The normality of more commonly used irrational numbers is an

open question--normality is actually a very hard property to prove for numbers

not designed specifically for the purpose of being normal.

+--First Church of Briantology--Order of the Holy Quaternion--+

| A mathematician is a device for turning coffee into |

| theorems. -Paul Erdos |

+-------------------------------------------------------------+

| Jake Wildstrom |

+-------------------------------------------------------------+

Aug 25, 1999, 3:00:00 AM8/25/99

to

Jake Wildstrom <wil...@mit.edu> wrote ...

: Daniel Pilloff <pil...@aecom.yu.edu> wrote:

: >Hi, what does normal mean?

:

: A number (in particular an irrational number, since

: rational numbers never have this property) is normal

: in base b iff in the base b expansion of the number,

: every finite digit sequence appears at least [once].

: Daniel Pilloff <pil...@aecom.yu.edu> wrote:

: >Hi, what does normal mean?

:

: A number (in particular an irrational number, since

: rational numbers never have this property) is normal

: in base b iff in the base b expansion of the number,

Normality in base b requires much more than that;

specifically, each sequence of n digits (n=1,2,...)

must have the limiting relative frequency 1/b^n.

--

r.e.s.

rs...@mindspring.com

Aug 25, 1999, 3:00:00 AM8/25/99

to

> The normality of more commonly used irrational numbers is an

> open question--normality is actually a very hard property to prove

> for numbers not designed specifically for the purpose of being normal.

Are there any results on density of normal irrationals? It's

interesting to find new (to me) dense sets of numbers that are

difficult to actually identify.

- James

Sent via Deja.com http://www.deja.com/

Share what you know. Learn what you don't.

Aug 25, 1999, 3:00:00 AM8/25/99

to

fo...@uq.net.au (Warren Shallcross) wrote, in part:

>Do you know of anyway of generating sequences of random numbers in your head

>(without becoming predictable - ie no repeating patterns)?

Quite difficult. I suppose you could try doing chain addition (which

isn't cryptosecure), which is sort of like a crude shift register:

start with a four digit number, add the first two digits mod 10, then

cross off the first digit and append the sum as the last digit.

But *no* method of generating a long sequence is likely to be carried

out perfectly in your head for that long...although that might just

help the randomness.

John Savard ( teneerf<- )

http://www.ecn.ab.ca/~jsavard/crypto.htm

Aug 26, 1999, 3:00:00 AM8/26/99

to

In article <7q1g71$9s1$1...@nnrp1.deja.com>, hol...@my-deja.com wrote:

=> Are there any results on density of normal irrationals? It's

=> interesting to find new (to me) dense sets of numbers that are

=> difficult to actually identify.

Normal numbers (all of which are irrational, of course) are much more

than just dense; their complement has measure zero.

One might say it is easier to find a needle in this haystack

than it is to find a bit of the hay.

Gerry Myerson (ge...@mpce.mq.edu.au)

Aug 26, 1999, 3:00:00 AM8/26/99

to

Gerry Myerson wrote:

> Normal numbers (all of which are irrational, of course) are much more

> than just dense; their complement has measure zero.

>

> One might say it is easier to find a needle in this haystack

> than it is to find a bit of the hay.

Interestingly, though, in the sense of category it's the other

way around: Normal numbers are conull as you point out, but they're

also meager.

At first glance this is counterintuitive. You might say, "but then (in base 10 e.g.)

what value can the frequency of 7's take on a non-meager set, if not 1/10"?

The answer is that for almost all numbers (in the sense of category), the frequency

of 7's does not exist at all. On a comeager set of reals, for any value r in [0,1] and

any epsilon, there will be infinitely many places in the decimal expansion of the

chosen real such that the frequency of 7's up to that spot is within epsilon of r.

Aug 30, 1999, 3:00:00 AM8/30/99

to

It's impossible. It can't be done. There is no way you or anyone else

can generate anything that is random. Weather it is inside your head or

not. There is no such thing as "Random". The smarter you are the more

random things you understand. This is called intelligence.

can generate anything that is random. Weather it is inside your head or

not. There is no such thing as "Random". The smarter you are the more

random things you understand. This is called intelligence.

Aug 31, 1999, 3:00:00 AM8/31/99

to MILKY WAY

On Mon, 30 Aug 1999, MILKY WAY wrote:

> There is no such thing as "Random".

The result of any experiment in which quantum mechanical effects have a

significant role.

Sep 2, 1999, 3:00:00 AM9/2/99

to

In article <7ptmgj$jue$2...@bunyip.cc.uq.edu.au>,

Warren Shallcross <fo...@uq.net.au> wrote:

)Hello all.

)

)Do you know of anyway of generating sequences of random numbers in your head

Warren Shallcross <fo...@uq.net.au> wrote:

)Hello all.

)

)Do you know of anyway of generating sequences of random numbers in your head

Do you mean "random" or do you mean "pseudorandom"? For the first, there

is no known way.

--

----

char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}

This message made from 100% recycled bits.

I don't speak for Alcatel <- They make me say that.

Sep 2, 1999, 3:00:00 AM9/2/99

to

How can you prove that a number is normal? I thought about it for a

while, but I just wouldn't know where to start from.

Can anyone make an example?

Thank you,

damio :D

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Sep 3, 1999, 3:00:00 AM9/3/99

to

In article <37CE40B4...@tiscalinet.it>, Damiano Mazza

<da...@tiscalinet.it> wrote:

<da...@tiscalinet.it> wrote:

> How can you prove that a number is normal? I thought about it for a

> while, but I just wouldn't know where to start from.

> Can anyone make an example?

The first & simplest example is Champernowne's number, formed by

concatenating the numbers 1, 2, 3, etc, like so:

.123456789101112131415161718192021222324252627282930313233....

The proof of its normality is not super-hard but not trivial, either.

Why not take a crack at it?

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