Patterns of Prediction

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Re: Patterns of Prediction

Post by satsuma »

RogerE wrote:
15 Feb 2021 01:38
Thanks Richard = capetriangle. I'm glad you enjoyed that pattern discussion that I headed
It started with Pythagoras.

Looking back, I notice I mistakenly referred to Equation (2) at one point, when I meant Equation (3).
My intention was to say


Here's another viewpoint for thinking about Equation (1) and Equation (3) in that post.

The sequence of squares of positive integers begins
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, ...
If you look at the gaps between pairs of numbers in that sequence, an interesting pattern emerges.
In particular, some gaps are equal to numbers which appear in the sequence itself. For example,
the gap between 16 and 25 is 9, which is also in the sequence, so 16 + 9 = 25 is a solution to Equation (1).
Again, the gap between 64 and 100 is 36, so 64 + 36 = 100 is another solution to Equation (1). (It is just
a scaled version of the previous solution, found by multiplying the three smaller numbers by 4.)
The gap between 144 and 169 is 25, so 144 + 25 = 169 is another solution to Equation (1).
Perhaps you will see that the gap between 225 and 289 is 64, so 225 + 64 = 289 also satisfies Equation (1).
The sequence of squares of positive integers begins
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, ...
.

Surely you are overthinking this sequence.

The increase between members of the series follows the very much simpler series:
3, 5, 7, 9, 11, 13, 15, 17, 19, ........
So every odd number in the sequence of squares will become a difference between two later numbers of the sequence!

The 9+16+25 and the 36+64=100 are just specific examples of Pythagoras right triangles.

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Re: Patterns of Prediction

Post by RogerE »

Not "overthinking", satsuma, just leading you (the reader) to see/discover the underlying pattern
for yourself. The most satisfying learning is not simply to memorise what you have been told, but
to reason out the result for yourself... I was leading you to see a proof that there are infinitely many
Pythagorean triples. Once you have seen it for yourself, it is satisfyingly "obvious", isn't it?
I wrote:If you look at the gaps between pairs of numbers in that sequence, an interesting pattern emerges.
Once you have seen that the first difference sequence of the sequence of squares is the sequence of
odd numbers, you can then see that every odd square is the difference between two consecutive squares,
providing a Pythagorean triple in which two members are consecutive integers.

It is a small further step to see that any even square is the sum of two consecutive odd numbers, so each
even square is the difference between two alternate squares, giving another family of Pythagorean triples.
For example, the even square 64 is the sum of two consecutive odd numbers 31+33; also 31 is the difference
between the consecutive squares 225 and 256, and 33 is the difference between the consecutive squares
256 and 289. Therefore 64 is the difference between the alternate squares 225 and 289. This gives us the
Pythagorean triple 8, 15, 17. It generalises to give Pythagorean triples in which two members differ by 2.
And so on.

By the way, if you calculate the second difference sequence of the sequence of cubes you will be able
to see that it also follows a "very much simpler" pattern. But it takes a more sophisticated argument from
there to show that there are no two positive cubes with sum equal to a cube.
Furthermore, if you calculate the second difference sequence of the sequence of squares, and the third
difference sequence
of the sequence of cubes, you will find even simpler patterns. This generalises.

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Figurate numbers

Let's discuss more patterns and ideas following the recent posts about square and cubes.

Appropriate numbers of dots can be arranged into geometric configurations.
Such numbers (of dots) are sometimes collectively called figurate numbers.

Squares and cubes

For example, 9 dots can be arranged into a 3 x 3 square configuration, so 9 is a
figurate number. Again, 64 dots can be arranged into an 8 x 8 square configuration,
so 64 is a figurate number. In the same way every number that is a "perfect square"
is a figurate number, since that many dots can be arranged into a square configuration.

Summary:
The (numerical) squares are figurate numbers — they correspond to (geometrical) squares.

Similarly, 8 dots can be arranged into a 2 x 2 x 2 cubic configuration (think of them occupying the
corners of a cube), so 8 is a figurate number. Again, 64 dots can be arranged into a 4 x 4 x 4 cubic
configuration, so 64 is a figurate number. In the same way every number that is a "perfect cube"
is a figurate number, since that many dots can be arranged into a cubic configuration.

Summary:
The (numerical) cubes are figurate numbers — they correspond to (geometrical) cubes.

Note that 64 is a figurate number in two different ways, as a square and as a cube. It is a perfect
sixth power, the condition for being both a square and a cube. Example: 729 = 27 x 27 = 9 x 9 x 9.

Summary:
There are infinitely many numbers that are both squares and cubes.

Triangular numbers

Now consider some different geometrical configurations. Evidently 3 dots can be arranged into an
equilateral triangle configuration, so 3 is a triangular number. Again, 6 dots can be arranged into an
equilateral triangle configuration (rows 1, 2, 3), so 6 is a triangular number. The sequence of triangular
numbers
begins
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, ...

.
download.png
.
https://plus.maths.org/content/maths-minute-triangular-numbers
.
This is another sequence of figurate numbers. Putting two consecutive triangular configurations
together forms a square, so the sum of any pair of consecutive triangular numbers is a square.
For example, 21 + 28 = 49 and 28 + 36 = 64. Note that 36 is a figurate number in two different
ways, as a triangular number and a square.

The first difference sequence of the sequence of triangular numbers is simply the sequence of
consecutive positive integers. When a positive integer is a triangular number, it is the difference
between two consecutive triangular numbers. For example, the triangular number 10 is the difference
between the consecutive triangular numbers 45 and 55, and therefore the triangular number 55 is the
sum of the triangular numbers 45 + 10. This generalises: there are infinitely many pairs of triangular
numbers with sum equal to a triangular number.

Exercise 1: Which pair of triangular numbers has sum equal to the triangular number 120?
Exercise 2: Which triangular numbers b and c satisfy 21 + b = c?
Exercise 3: After 36, what is the next triangular number which is also a perfect square?

Summary: It can be proved that
For any positive integer n, there are triangular numbers b and c which satisfy n + b = c
Summary: It can be proved that
There are infinitely many triangular numbers which are also squares.

Tetrahedral numbers

Just as cubes are the three dimensional analogues of square configurations (made by stacking squares),
the three dimensional analogues of triangular configurations are tetrahedral configurations (made by
stacking triangular configurations.

Rather than configurations of dots, it is more physically realistic to think of configurations of spheres,
say oranges. Three oranges form a triangular configuration, and one more orange placed on top forms
a tetrahedron = triangular pyramid. Thus 1 + 3 = 4 is a tetrahedral number.

560px-Tetrahedron.jpg
.
Wikipedia wrote:In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces
https://en.wikipedia.org/wiki/Tetrahedron
.
download.jpg
.
https://www.physicsforums.com/threads/close-packing-of-spher ... -c.771476/
.
Again, 6 oranges can be arranged into a triangular configuration, with 3 placed as a second layer (in
the gaps between the oranges in the first layer), and one more orange added as a third layer. This
stack is a tetrahedral configuration, of 1 + 3 + 6 = 10 oranges, so 10 is a tetrahedral number. The
sequence of tetrahedral numbers begins
1, 4, 10, 20, 35, 56, 84, 120, 165, ...

.
The first difference sequence of the sequence of tetrahedral numbers is the sequence of
triangular numbers. Notice that 10 is both a triangular number and a tetrahedral number. The
same is true for 120.

Exercise 4: Are there infinitely many pairs of tetrahedral numbers with sum equal to a
tetrahedral number?
Exercise 5: Are there infinitely many tetrahedral numbers that are also triangular numbers?
.
Stacked canon balls — Italian military scrip
Stacked canon balls — Italian military scrip
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Mnemonics

The word mnemonic derives from Greek μνήμη [mnēmē] — remembrance, memory.
It is directly related to Greek μνημονικός [mnēmonikos] — of memory, relating to memory.

Some recent posts in this thread have presented mnemonics. In particular,
posts have presented mnemonics for the first eight decimal places of pi (π), and for
the location of letters in a printer's California Job Case. Let's think about how/why
a mnemonic works.

A mnemonic is usually a way of remembering a particular list by using a pattern
which encodes the list but is "easier" to remember because of its structure.

Here are some familiar examples:

Mnemonic for the order of colours in the spectrum or rainbow:

The made-up name "Roy G. Biv" is an acronym with enough structure for us to memorise
it, whereas the list
.
red, orange, yellow, green, blue, indigo, violet
.
has no obvious structure to help us remember it directly.

Mnemonic for the order of the Great Lakes on the Canada/USA border:

The sentence "See Mr Harris eating navel oranges" has enough structure for us to memorise
it, and its initials prompt us to reconstruct the sequence
.
Superior, Michigan, Huron, Erie, Niagara, Ontario
.
That version locates the Niagara Falls in the appropriate part of the sequence.

Mnemonic for the order of the planets in the Solar System:

The sentence "My very educated mother just showed us nine planets" has enough structure for us
to memorise it, and its initials prompt us to reconstruct the sequence
.
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, [Pluto]
.
That version locates Pluto in its proper position. the only question is whether to still mention it,
since it has undergone a downgrade of status to "dwarf planet".

Mnemonic for the scientific classification used in taxonomy:

The sentence "Kings play chess on funny glass stairs " has enough structure for us
to memorise it, and its initials prompt us to reconstruct the sequence
.
Kingdom, Phylum, Class, Order, Family, Genus, Species
.
Classification of mnemonics:

A careful study of more examples of mnemonics suggests a classification into various types.
The Roy G. Biv example is an acronym mnemonic. The other examples listed above
can be called phrase mnemonics, where the initial letter of each word is the first
letter of each word in the sequence represented.

A body mnemonic uses parts of the body to encode a sequence to be memorised.
My favourite example is this mnemonic for the number of days in each month of the
Gregorian calendar:

.
.<br />Body mnemonic for number of days in each month<br />Two fists:<br />Tops of knuckles are 31 day months<br />Gaps between knuckles are 30 day months (or February!)<br />Notice how July and August are accommodated
.
Body mnemonic for number of days in each month
Two fists:
Tops of knuckles are 31 day months
Gaps between knuckles are 30 day months (or February!)
Notice how July and August are accommodated
.
My main source for information in this post is
https://en.wikipedia.org/wiki/Mnemonic
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Serendipity

Serendipity is our "explanation" of the happy feeling we get when two seemingly unrelated events
happen close together, and we perceive a positive relationship between them. There are other terms
which are variously applied to such circumstances including: fluke, lucky break, happy chance, accidental
discovery, stroke of good luck.

There are also cases in which we perceive a negative relationship between two seemingly unrelated events.
Terms such as misfortune and stroke of bad luck are used in such circumstances.

I suggest that these are instances of pattern perception. Each day there are many events in our lives. In
typical cases we perceive cause-and-effect relationships ("dependence"), and in other cases we perceive
no relationship ("independence"). In the latter class, we normally perceive no pattern, but in the sheer huge
number of such events there will sometimes be an "overlap" that we perceive as a linkage. It is just a matter
of probability. "Given enough instances, whatever can happen will happen". When our emotional response is
positive we speak of serendipity or stroke of good luck. When our emotional response is negative we speak
of misfortune or stroke of bad luck.
.. o O O O o ..
.
I recall a "serendipity" post by much-missed member RobRoyH, who was last active on Stampboards in
Sept 2020. "Serendipity" was a recurrent theme in his posts. This particular post was in the thread about
Irish SOAR stamps (SOAR = "Stamps on a Roll").
RobRoyH wrote:
21 Apr 2020 17:14
Hopefully this posting will not be unwelcome. I am currently processing an International Society of Worldwide Stamp Collectors swap packet. It is SUPPOSED to be made up of all United States material. This poor vandalized item shows up.... where have I seen this before?

Image
2010 Ireland Christmas SOAR Stamp - Left Half Only!

No doubt some well meaning church lady cut it in half, keeping only the pretty picture, and dutifully deposited it in the mission box along with the other three stamps she had seen since the last Sunday service. In the intervening decade, it has traveled to some kiloware lover in the US who no doubt did not know what to make of it.

Only someone who is following this spectacular thread [on Irish SOARs] with baited breath could POSSIBLY have recognized what exactly it is!

I call that SERENDIPITY, possibly even a minor miracle! Another nail in the coffin for that old dead philosophy where coincidence and random chance are given credence.

Regular readers of my inconsequential threads will recall how much pleasure I take in such occurrences!

Under normal circumstances, I would probably throw it in the wounded solder envelope for use in some future Stamp Art project.

For the time being, I will mount it up with my other three SOARS and continue following this spectacular thread to see if the secret and hidden rulers of the universe have any more messages to channel through our most excellent and illustrious Rein!
A superstitious mindset might venture to assign supernatural causes to such events, because we do
like to find an "explanation" (to give us influence/control over what happens, and the power to predict).
The purely objective, rational explanation is that it's simply probability. That explanation does not give
us influence/control over what happens, so it is less seductive, and the only sense in which it gives us the
power to predict is in the statistical sense ("the probability of this happening is p", where p is a number
which we might be able to calculate or estimate, but in many cases we have no idea what number p is!).
Rationality requires us to give up the wish to control in those circumstances where no mechanism for
control actually exists.

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Date Patterns

Noting date patterns depends on the dominant date notation in local use.

Today is 4321 = 4 March 2021 [Thursday]

Later this month Pi Day = 314 = March 14 [a Sunday in 2021]

US: Next month 4321 = April 3, 2021 [Saturday]

US: The following month Star Wars Day = 54 = May the 4th [a Tuesday in 2021]

Later this year 192021 = 1 Sept 2021 [Wednesday]
(Alternatively, in US date convention: Saturday, Jan 9, 2021)

A palindromic date this month: 12321 = 12 March 2021 [Friday]
In US date convention, depending on how you parse the digit string:
12321 = Jan 23, 2021 [Saturday]
12321 = Dec 3, 2021 [Friday]

A rep-digit date for next year:
22222 = 22 Feb 2022 [Tuesday]
US convention: 22222 = Feb 22, 2022 [Tuesday]

/RogerE :D

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Re: Patterns of Prediction

Post by castores »

A most interesting thread and pretty much beyond me.

Amazing and keep the crab flowing (I will try) :D
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Re: Patterns of Prediction

Post by castores »

I would just like to show I have some understanding.

My current post replies is 1743 and with this one it will be 1744.

That is n+1 (n being = to 1743 :lol: :lol: :lol: )

How good is that?

(note I went to Calc 4 at College, never went to a class (thanks to my teacher of 4 years who knew where I was at) and came top of the class.

Today, I have no idea of "d over t" or any of that :oops:

(Peter Enge, great teacher, great person)

(ps. I had to take two courses at the same time, the other was film genre and you really couldn't miss the movie - no chance of catching it later - watched some great movies)

(spent too much time playing cards in the cafeteria)
(once we left school first thing in the morning and drove to the nearest snow, we go back just in time for students to be leaving and we had buckets full of snow - and pelted everyone :lol: :lol: :lol:
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Re: Patterns of Prediction

Post by RogerE »

Hello castores. First a brief mathematics tutorial (to fit the theme of this thread),
followed by a few remarks about nomenclature ;)

Mathematics tutorial

The prime numbers are the elements from which all positive integers are built (by multiplication).
For example, 1743 = 3 x 7 x 83 and 1744 = 2 x 2 x 2 x 2 x 109.
Beginning with 4 = 2 x 2, the whole numbers [positive integers] that are not primes
are called composite numbers = numbers composed of primes.
If a positive integer [whole number] is not a prime, it must be divisible by a prime no larger than its
square root (because the product of two numbers greater than its square root would have to be larger
than the number itself).

The list of prime numbers begins 2, 3, 5, 7, 11, 13, 17, 19, ...
There are 25 primes below 100 ("a quarter of the first hundred numbers are primes").
But they thin out, though the detailed spacing is irregular.
Does the list of primes end? The classical Greek mathematician Euclid of Alexandria (flourished
around 300 BCE) produced a conclusive proof that there is no largest prime number (so the list
never ends). Here's how the proof goes. I'll split it into two parts.

First part: Suppose a and b are two numbers which are both multiples of a particular prime
number p. Then they differ by a multiple of p, and 2 is the smallest prime, so the difference
between a and b is at least 2. This means that any two consecutive numbers cannot both
be divisible by the same prime number, since they only differ by 1.

Second part: Suppose there is a largest prime, P. We will show that this assumption
leads to a contradiction, and therefore the assumption itself must be false. Multiply together all
the prime numbers 2, 3, 5, ..., P. This yields a large composite number, which we can denote
by C. Now consider the next number, C+1. By the first part of the argument, C+1
cannot be a multiple of any of the primes 2, 3, 5, ..., P, because they are all factors of C.
Therefore every prime factor of C+1 must be greater than P. But that contradicts the
assumption that P is the largest prime number. By contradiction, there cannot be a largest
prime number, so the list goes on endlessly.

(Note that in this argument C+1 could turn out to be prime or composite, it doesn't matter
which. All that is important is that it must have at least one prime factor, possibly itself, and any
such prime factor must be greater than P.)
.
Maldives, 1988: Euclid of Alexandria
Maldives, 1988: Euclid of Alexandria
.
If you think about it, Euclid's argument shows that if N is the product of any given collection X
of primes, then every prime factor of N+1 lies outside the collection X. For example, suppose
X is the collection {2, 3, 3, 5}, repetition being allowed. Then N = 2 x 3 x 3 x 5 = 90, so all
prime factors of N+1 must lie outside the collection {2, 3, 3, 5}. In fact, N+1 = 91 = 7 x 13,
and the collection {7, 13} has no overlap with {2, 3, 3, 5}.

To take one more example, if X is {2, 2, 7, 13} the product is N = 2 x 2 x 7 x 13 = 364. Then
N+1 = 365 = 5 x 73 and {5, 73} has no overlap with {2, 2, 7, 13}. Incidentally, that tells you
something about the number of days in a year!
_____________
.
Nomenclature (for castores followers)

Here is a piece of information castores probably knows well, but other Stampboarders might find it informative:
The Free Dictionary by Farlex wrote:Castor:
1. Greek Mythology: One of the Dioscuri [See the next excerpt, from Wikipedia].
2. A binary star in the constellation Gemini, approximately 46 light years from Earth and of roughly equal brightness with Pollux.
[Latin, from Greek Kastōr, twin of Pollux.]
https://www.thefreedictionary.com/Castores#:~:text=1.,Kast%C ... Pollux.%5D
.
Further clarification, plus a noteworthy two-word phrase, comprising a total of 30 letters :D
Wikipedia wrote:Castor and Pollux were twin half-brothers in Greek and Roman mythology, known together as the Dioscuri. Their mother was Leda, but they had different fathers... The pair are thus an example of heteropaternal superfecundation. {I couldn't resist sharing that phrase!]

In Latin the twins are also known as the Gemini (literally "twins") or Castores.
https://en.wikipedia.org/wiki/Castor_and_Pollux
.
Italy, 1983: Commemorative cover and pictorial cancel<br />celebrating Castor and Pollux.
Italy, 1983: Commemorative cover and pictorial cancel
celebrating Castor and Pollux.
.
/RogerE :D

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Re: Patterns of Prediction

Post by castores »

Thank you RogerE,

A most interesting read which I actually followed.
Mind I don't think I could explain it after one read :lol:

castores

Ah yes, in deciding on a moniker I toyed with all options regarding Castor and Pollux but decided the title combining the two looked best.
The pair are thus an example of heteropaternal superfecundation.
Superfecundah !

Castor is considered the child of King Tyndareus, whilst Pollux the offspring of Zeus.
As such Castor was mortal.

Thanks RogerE, I will place a post in my "duplicates" thread regarding this - hadn't thought to share it :oops: and I don't want to clutter your thread.

caspol.jpg
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Re: Patterns of Prediction

Post by RogerE »

There, you see, castores: although you were a self-proclaimed mathematical philistine, a
clear and enthusiastic presentation of some mathematical reasoning nevertheless allowed
you to understand and enjoy it. One experience like that can help to cancel out some poor
past experiences, and bring on the joy of understanding and the sense of inherent beauty.

Symmetry in Roman numeral notation

Some mathematics I was doing a few hours ago made me think about Roman numerals, and
the characters used to represent them: I, V, X, L, C, D, M. Four of these seven characters have a
vertical axis of symmetry — the right half of the character is the mirror image of the left half.
For brevity, let's call those the mirror characters. So {I, V, X, M} is the set of mirror characters
which appear in Roman numerals.

Let's think about strings ("lists") of these mirror characters, without worrying whether they are
actual valid Roman numerals.

A whole string of mirror characters can have a vertical axis of symmetry through the middle of the
string, if the right half of the string is the mirror image of the left half. Again for brevity, let's call
any such string of characters a mirror string. Here is how the list of mirror strings begins:
I, V, X, M
II, VV, XX, MM
III, IVI, IXI, IMI, VIV, VVV, VXV, VMV, XIX, XVX, XXX, XMX, MIM, MVM, MXM, MMM
Suddenly, when we get to strings of three characters, the number of possibilities mushrooms!

So, here is my mathematical challenge for castores. (Other Stampboarders might also like to
try it, but please don't post your "answers" or "comments" yet. Let's see if castores can continue
to build on his recent mathematical enjoyment.)

Let S(n) be the number of mirror strings of length n, that is, formed from n mirror characters,
repetition of characters being permitted. We have seen that
S(1) = 4, S(2) = 4, S(3) = 16.
What number is S(4)? What is S(5)? Can you find a pattern that would allow you to predict S(n) for
any n greater than 3?

/RogerE :D

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Re: Patterns of Prediction

Post by castores »

RogerE wrote:
07 Mar 2021 03:33
There, you see, castores: although you were a self-proclaimed mathematical philistine, a
clear and enthusiastic presentation of some mathematical reasoning nevertheless allowed
you to understand and enjoy it. One experience like that can help to cancel out some poor
past experiences, and bring on the joy of understanding and the sense of inherent beauty.

Symmetry in Roman numeral notation

Some mathematics I was doing a few hours ago made me think about Roman numerals, and
the characters used to represent them: I, V, X, L, C, D, M. Four of these seven characters have a
vertical axis of symmetry — the right half of the character is the mirror image of the left half.
For brevity, let's call those the mirror characters. So {I, V, X, M} is the set of mirror characters
which appear in Roman numerals.

Let's think about strings ("lists") of these mirror characters, without worrying whether they are
actual valid Roman numerals.

A whole string of mirror characters can have a vertical axis of symmetry through the middle of the
string, if the right half of the string is the mirror image of the left half. Again for brevity, let's call
any such string of characters a mirror string. Here is how the list of mirror strings begins:
I, V, X, M
II, VV, XX, MM
III, IVI, IXI, IMI, VIV, VVV, VXV, VMV, XIX, XVX, XXX, XMX, MIM, MVM, MXM, MMM
Suddenly, when we get to strings of three characters, the number of possibilities mushrooms!

So, here is my mathematical challenge for castores. (Other Stampboarders might also like to
try it, but please don't post your "answers" or "comments" yet. Let's see if castores can continue
to build on his recent mathematical enjoyment.)

Let S(n) be the number of mirror strings of length n, that is, formed from n mirror characters,
repetition of characters being permitted. We have seen that
S(1) = 4, S(2) = 4, S(3) = 16.
What number is S(4)? What is S(5)? Can you find a pattern that would allow you to predict S(n) for
any n greater than 3?

/RogerE :D
I have to say RogerE, I have only just read the above and am very intrigued!

Currently enjoying the outside and the birds that abound so I will not spend any time (unless my mind wanders - for those are times when it happens - an empty mind allows many tangents to appear without coercion.

So maybe things 'will appear' or I will just enjoy the scenery... but I will get to this - I like it!

I do wonder though, in terms of your nomenclature (is that correct, english was never my best at school, so I'm probably incorrect) "mirror strings" - is that your own or gleaned from elsewhere?

Roman numerals have always been of interest.

I will get back to you RogerE.... :)
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Re: Patterns of Prediction

Post by castores »

Still spending time outside but I did consider something for a moment and I have to say it wasn't in the way I thought it would be.

It is about this:
I, V, X, M
II, VV, XX, MM
III, IVI, IXI, IMI, VIV, VVV, VXV, VMV, XIX, XVX, XXX, XMX, MIM, MVM, MXM, MMM
RogerE, I have never heard of IVI (let alone VV, etc) used in Roman Numerals, unless all I understand about them can be thrown out the window.

LXVI = 66 (read backwards is 66, but not in RN)

(one of the best things about old movies MCMLXI)

All that aside, I will still see if I can make sense of and try to answer your conundrum.

Just for fun:

Is VV = V + V (10 or X) or V x V (25 or XXV) or V - V (0 or ?) [less likely] or V / V [even less likely).
{{nothing like some bad grammar}}

XX = 20

IVI = some sort of intravenous drip?

VVV = straight out of MSTK3000
(obviously VVVBMB = vavavaboomb)

MMM = nice......

MSTKMMM = crappy b-grade movies that got an overdub of two dudes taken the pi** out of it.

(MSTK3000 = Mystery Science Theatre 3000 - nothing to do with any of the above :D :lol: :D :lol: )

Ps. afterthought, some things no matter how interesting can become tedious and hard to follow, may I suggest some humour (in thought) to keep it rolling - Maybe some collaboration? (Who's the straight guy? - maybe bazza could join us :lol: )

Considered: information given and non-conforming Roman Numerals I'm not sure of your S(n)....

Edit

Maybe this is where I lost the plot?
and
the characters used to represent them:
"characters used to represent them"
I think maybe I should revisit.... :roll:

I'm thinking after realising that sentence I've wasted much space RogerE :oops:
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Re: Patterns of Prediction

Post by castores »

It does still bring forth the question (mind you the actual question hasn't even been thought about :lol: )
If I am to use the Roman Numerals (in whatever format I like), your list isn't numerical in the RN sense so how do I use it numerically (as suggested by your list - sort of) BUT the numerals aren't actually numerically 'sensible' for math numeracy I know of. (wow, that's a lot of 'nume')

So, without this being explained, I can't progress.

For example, I could say your listing is missing of numbers.
You have MXM, MMM
But why isn't (eg) MXIIXM in between? (very pretty, makes me think of Maxim or Marxism).

Possibly that could read 1912 but I'd prefer to say MCMXII.

(Ps. I'm expecting that you RogerE will show me the errors of my ways... A fundamental misstep along the way perhaps?)
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Re: Patterns of Prediction

Post by satsuma »

I suspect that with 4 digits the possibilities will be the same as with 3.

Further as the number of digits increases there will be growth with odd numbers and no growth with even digits.

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Re: Patterns of Prediction

Post by castores »

satsuma wrote:
07 Mar 2021 19:49
I suspect that with 4 digits the possibilities will be the same as with 3.

Further as the number of digits increases there will be growth with odd numbers and no growth with even digits.
But the increase in numbers that could be used would increase greatly.
Quite the reverse of something like prime numbers.
Any correlation? Just asking?

(again, maybe I'm plotless)

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Re: Patterns of Prediction

Post by RogerE »

Hello castores. I think it might help if you reread my post about mirror characters among the set of characters used in Roman numerals. I went to some effort to make it clear and user-friendly. I think
you might have read it quickly, and skipped some of the details, though I did write it particularly to be
helpful for you.

To answer one of your questions, this passage shows that I was introducing the terminology rather than citing terminology that has been standardised:
For brevity, let's call those the mirror characters. So {I, V, X, M} is the set of mirror characters
which appear in Roman numerals.


A key sentence in my "invitation post" is this:
Let's think about strings ("lists") of these mirror characters, without worrying whether they are
actual valid Roman numerals.


A character string is just a list of characters, it doesn't have to carry any "meaning". For example, VV and IVI
are examples of mirror strings from this character set, even though these strings are not standard Roman numerals. It happens that the mirror strings III, XIX and MXM are actual Roman numerals, but there are very few such examples.

So, after rereading my "invitation post", reassure yourself that
S(1) = 4, S(2) = 4, S(3) = 16.
makes sense and you can see that it's correct. Then tackle my suggested exercises:
• What number is S(4)?
• What is S(5)?
• Can you find a pattern that would allow you to predict S(n) for any n greater than 3?

Your friend,
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Interestingly, I just decided to "tweak" that last post, and the "pencil icon" in the top right let me edit it,
but it wouldn't let me save the changes!

I made a copy if what they were going to be, "just in case". Here they are:

A couple of longer mirror strings, just to broaden your perspective, are XXXXVXXXX and MIIMMIIM.

So, after rereading my "invitation post", reassure yourself that
S(1) = 4, S(2) = 4, S(3) = 16.

makes sense and you can see that it's correct. Then tackle my suggested exercises:
• What number is S(4)?
• What is S(5)?
• Can you find a pattern that would allow you to predict S(n) for any n greater than 3?

Your friend/friendly tutor, :)
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Catch-up comments for castores and satsuma:

• For castores:
castores wrote:For example, I could say your listing is missing of numbers.
You have MXM, MMM. But why isn't (eg) MXIIXM in between?
Answer: because the strings of characters are being listed by length.
MXM, MMM are both strings of length 3, whereas MXIIXM has length 6,
and won't appear in the sequence until all mirror strings of length 4
have appeared, then all mirror strings of length 5. Then quite a few mirror
strings of length 6 must also appear (IIIIII, IIVVII, IIXXII, IIMMII, IVIIVI,... )
before it is the turn for MXIIXM.

Remember, here we are just studying the mirror patterns. Any possible "meaning"
for these strings is irrelevant to the order in which they appear.

• For satsuma:

Slightly tweaked,
satsuma wrote:
07 Mar 2021 19:49
I suspect that with 4 digits [characters] the possibilities will be the same as with 3.
Further as the number of digits [characters] increases there will be growth with
odd numbers [lengths] and no growth with even digits [lengths].
Yes satsuma, you are correct!
Can you (or castores) say how much growth occurs when the sequence passes
from all patterns of a particular even length, to all patterns of the next odd length?
Can you state it compactly as a formula involving S(n)?
________________
.
Thanks to both of you for thinking about these things. Much of Mathematics is about
the study of pattern (not just "doing sums"!), so these discussions are "mathematical".

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

I think it's time that I posted some more about mirror strings of characters from the set {I, V, X, M}.
Informally, these are "mirror strings of characters used in Roman numerals".

The list of mirror strings begins
I, V, X, M
II, VV, XX, MM
III, IVI, IXI, IMI, VIV, VVV, VXV, VMV, XIX, XVX, XXX, XMX, MIM, MVM, MXM, MMM
....
Using S(n) to denote the number of strings of length n, we see that S(1) = 4, S(2) = 4, S(3) = 16.

It has been commented by satsuma that the number of mirror strings of a given even length
is equal to the number of mirror strings of the preceding odd length. Therefore S(4) = 16.

Let W be any particular mirror string of length n. If x is any one of the four characters in the set {I, V, X, M},
then xWx will be a mirror string of length n + 2.
For instance if n = 3, and W is the string VMV, the strings of length 5 with W as their "interior part" are
IVMVI, VVMVV, XVMVX, MVMVM
Any particular W of length n is the "interior part" of four strings of length n + 2. Therefore
S(n + 2) = 4 x S(n)
Since S(1) = 4, it follows that S(3) = 4 x 1 = 4, S(5) = 4 x 4 = 16, S(7) = 4 x 16 = 64, S(9) = 4 x 64 = 256, ...
Since S(2) = 4, it follows that S(4) = 4 x 1 = 4, S(6) = 4 x 4 = 16, S(8) = 4 x 16 = 64, S(10) = 4 x 64 = 256, ...
So, in principle, this settles the question How many mirror strings of length n are there?
I say "in principle", because for any particular n the solution so far is a procedure for calculating S(n)
from S(1) or S(2).

It is mathematically more satisfying if we can explicitly specify the number S(n), rather than giving it
implicitly as a procedure for calculating it from S(1) or S(2).

Any even number is of the form 2k, where k is a whole number [integer].
So, if k = 1 then S(2k) = S(2) = 4;
if k = 2 then S(2k) = S(4) = 4 x 4 = 16;
if k = 3 then S(2k) = S(6) = 4 x 4 x 4 = 64; etc,
The pattern is S(2k) = 4 x 4 x ... x 4 (a product of k factors 4) = "4 to the power k".
The odd number preceding 2k is 2k – 1.
So, if k = 1 then S(2k – 1) = S(1) = 4;
if k = 2 then S(2k – 1) = S(3) = 4 x 4 = 16;
if k = 3 then S(2k – 1) = S(5) = 4 x 4 x 4 = 64; etc,
The pattern is S(2k – 1) = 4 x 4 x ... x 4 (a product of k factors 4) = "4 to the power k".
One very useful, compact notation for "4 to the power k" is 4^k. (Another puts "k" as a tiny symbol at the top right corner of 4, but that "two level" notation is not as convenient as 4^k, where both symbols are the same size, and both sit on the same level.)

Summary: For any positive integer k, the number of "Roman" mirror strings of length n = 2k –1 or n = 2k is
S(2k – 1) = S(2k) = 4^k.
.
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

By the way, the list of powers of 4 begins
4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024,
4^6 = 4096, 4^7 = 16384, 4^8 = 65536, 4^9 = 262144, 4^10 = 1048576,
....
So, in the notation for counting "Roman" mirror strings used in recent posts in this thread,
S(2k-1) first exceeds a thousand when k = 5, and first exceeds a million when k = 10. In fact,
S(9) = 1024, S(19) = 1048576
.
Footnote: If you're "tuned in" to patterns in the powers of 4, you might like
4^10 = (4^5)^2 = 1024^2 = (1000 + 24)^2
= 1000^2 + 2 x 24 x 1000 + 24^2
= 1000000 + 48000 + 576
= 1048576

/RogerE :D

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Re: Patterns of Prediction

Post by Lakatoi 4 »

With many theorems above here can can someone point me to the one that predicts Saturday’s winning Lotto numbers, I’d even accept who wins the first race at Randwick on Saturday :D

Sorry for being facetious about this but while these theorems can correctly predict a massive number of things, they simply cannot factor in changes to the result where there are a huge number of complex and ever changeable variables involved.

So we are left with the only true constant that’s impossible to factor into any equation.......LUCK :!:
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Re: Patterns of Prediction

Post by Ubobo.R.O. »

I predict that the first race on Saturday will actually be run at Rosehill and will be won by Great House.
Full time horse non-whisperer, post box searcher and lichen covered granite rock percher. Gee I'm handsome !
You gottem birds, butterflies, shells, maps, flags and heads on stamps ? Me wantem !

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Re: Patterns of Prediction

Post by RogerE »

Well Lakatoi 4, let's rationally examine your comments.

There is more than one category of "prediction".

Deterministic prediction would be predicting an outcome for which all the governing factors
are known and invariant. For instance, predicting that the number of "Roman mirror strings"
of length 10 will turn out to be S(10) = 1024, is a deterministic prediction. I haven't tried to
write out the list, but I've discussed the governing factors, which are deterministic, and I have
applied my knowledge of them to calculate the outcome.

It was once thought that the physical universe is like a giant clock mechanism. If only you knew the
mechanical details and the position of all the cogs at one particular time, you would be able to predict
the detailed state of the universe at any future time. This mechanistic view has been assailed by
various philosophical and mathematical criticisms. It may be that certain physical processes are
inherently "probabilistic" rather than "deterministic". Even for deterministic processes, it may be
the case that their future "trajectories" depend so sensitively on the current state that two possible
current states, closer together than we can measure/perceive any difference, might still lead to vastly
different trajectories in a relatively short passage of time.

Statistical prediction would be predicting an outcome on the basis of probabilities. In this
case, a collection of instances of some "event" can be observed and "frequencies" of various outcomes
tabulated. If a many instances have been observed and counted into the tabulated "frequencies", the
reliability of the calculated probabilities is "high" (often quantifiable), even though the full set of
governing factors might not be well understood. The ability to predict the outcome of a particular
instance remains subject to uncertainty, even though the "probability" of a particular outcome is
quantifiable within a very narrow range. On the other hand, if the outcomes of a good number of
instances are to be predicted, the number of instances with a particular outcome within that group
can be predicted within a very narrow range. For example, insurance companies have very detailed
probability tables for a person's life expectancy. The actual age at which a particular individual will
die is scarcely a matter that those probability tables can forecast within any narrow range. On the
other hand, a cohort of 100 people can likely have the life expectancy distribution predicted relatively
rather tightly — how many will live to each possible age, but not which individuals will live to any possible
age. All this is not "luck" so much as the "cancelling out" of unusual factors over a large group of instances.
Bookmakers/totalisators are able to make a profit because of statistical factors: they attempt to use
past information/performances to set "odds" (probabilities) of outcomes, and over enough instances
the predictions are statistically accurate within increasingly narrow ranges. However, the bookmaker
adds a "loading" to the offered odds to ensure profitable outcomes "in the long run".

"Luck" may really be just a cover name for "a set of governing factors which include many unknown or poorly understood factors".

Psychological prediction would be predicting an outcome on the basis of known human beliefs,
preferences, tendencies and/or wishes. For example, predicting when your daughter will next telephone
you is likely to be more accurate than predicting when you will receive a phone call from a "distant" relative.

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Let's have another look at characters/glyphs with vertical mirror symmetry.
A character has vertical mirror symmetry if its right half is the mirror image of its left half.
If we take the capital letters of the English alphabet as our source of characters, the set of characters with vertical mirror symmetry is
{A, H, I, M, O, T, U, V, W, X, Y}.
For brevity, let's call them the "English mirror capitals". Obviously those eleven characters
include the four "Roman mirror characters".

A character string has vertical mirror symmetry if its right half is the mirror image of its
left half. Let E(n) be the number of mirror strings of English capital letters. The analysis in
earlier posts counting mirror strings of Roman characters easily adapts to showing that
E(1) = 11, E(2) = 11
E(n + 2) = 11 x E(n)
It follows, for all positive integers k, that
E(2k – 1) = E(2k) = 11^k.
Hence the number of mirror strings of length 5 exceeds a thousand, and the number of length 11 exceeds a million. In fact, E(5) = 1331 and E(11) = 1771561.

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

As a minor amusement, you might like to find some actual words which are strings with vertical mirror symmetry, composed from capital letters in the set {A, H, I, M, O, T, U, V, W, X, Y}.
Examples include
AHA, TOT, MUM, MA'AM, WOW

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Re: Patterns of Prediction

Post by castores »

RogerE, so it doesn't get lost in junk - I have sent you an email :)
Best regards,
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Re: Patterns of Prediction

Post by castores »

RogerE wrote:
11 Mar 2021 22:12
As a minor amusement, you might like to find some actual words which are strings with vertical mirror symmetry, composed from capital letters in the set {A, H, I, M, O, T, U, V, W, X, Y}.
Examples include
AHA, TOT, MUM, MA'AM, WOW
TIT, TUT, TOOT
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Re: Patterns of Prediction

Post by castores »

HAH

Hah!

eXTRA

WOW MUM, TOT.

MA'AM, TIT TUT TOOT...

TOOT TOOT

HAH

(well I thought it was funny)
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Re: Patterns of Prediction

Post by RogerE »

Shall we perhaps have one more look at characters/glyphs with vertical mirror symmetry?

A character has vertical mirror symmetry if its right half is the mirror image of its left half.
If we take the lower case letters of the font used by Stampboards as our source of characters, the set of characters with vertical mirror symmetry is
{i, l, o, v, w, x }
(Note that t does not quite have vertical mirror symmetry, though the corresponding letter in some plainer fonts does have vertical mirror symmetry.)

The Stampboards font also some pairs with vertical mirror symmetry:
(b, d), (p, q)
.
A character string has vertical mirror symmetry if its right half is the mirror image of its
left half. For instance, vlpdxvxbqlv is such a string. The list begins
i, l, o, v, w, x;
bd, db, ii, ll, oo, pq, qp, vv, ww, xx;
bid, bld, bod, bvd, bwd, bxd, dib, dlb, dob, dvb, dwb, dxb, iii, ili, ioi, ivi, iwi, ixi, ...
....
Let L(n) be the number of length n mirror strings of lower case Stampboards letters.
Then L(1) = 6, L(2) = 10. I have shown the first 18 mirror strings of length 3. You can check that L(3) = 60.

For any length n mirror string W of lower case Stampboards letters, the string zWz' is a mirror string of length n + 2 provided z is a letter from the set {i, l, o, v, w, x } and z' = z, or else z is a letter from the set {b, d, p, q} and z' is its vertical mirror image. Therefore
L(n + 2) = 10 x L(n)
.
From the cases L(1) = 6, L(2) = 10 we have
L(3) = 10 x L(1) = 10 x 6 = 60; L(5) = 10 x 60 = 600; L(7) = 10 x 600 = 6000; ...
L(4) = 10 x L(2) = 10 x 10 = 100; L(6) = 10 x 100 = 1000; L(8) = 10 x 1000 = 10,000; ...
.
It follows, for all positive integers k, that
L(2k – 1) = 6 x 10^(k – 1)
L(2k) = 10^k
Hence the number of vertical mirror strings of lower case Stampboards letters of length 12 is a million.

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

A friend has sent me a link for Pi Day, including this delicious-looking pie:

Photo credit: Oksana Mizina/Shuttercock
Photo credit: Oksana Mizina/Shuttercock
.
This image is in a blog in The Old Farmer's Almanac, apparently written by astronomer Bob Berman.
https://tinyurl.com/45hx8zyf

The whole blog is quite relevant to topics discussed in this Patterns of Prediction thread.
Pi Day was one of the topics in my 4 March post:
RogerE wrote:
04 Mar 2021 12:22
Date Patterns

Noting date patterns depends on the dominant date notation in local use.

Today is 4321 = 4 March 2021 [Thursday]

Later this month Pi Day = 314 = March 14 [a Sunday in 2021]

US: Next month 4321 = April 3, 2021 [Saturday]

US: The following month Star Wars Day = 54 = May the 4th [a Tuesday in 2021]

Later this year 192021 = 1 Sept 2021 [Wednesday]
(Alternatively, in US date convention: Saturday, Jan 9, 2021)

A palindromic date this month: 12321 = 12 March 2021 [Friday]
In US date convention, depending on how you parse the digit string:
12321 = Jan 23, 2021 [Saturday]
12321 = Dec 3, 2021 [Friday]

A rep-digit date for next year:
22222 = 22 Feb 2022 [Tuesday]
US convention: 22222 = Feb 22, 2022 [Tuesday]

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Ubobo.R.O. wrote:
10 Mar 2021 18:46
I predict that the first race on Saturday will actually be run at Rosehill and will be won by Great House.
The Ubobo.R.O. prediction was unsuccessful, and not surprisingly, he hasn't "fessed up" here.
In fact Great House placed fourth, 4.1 lengths behind the winner, out of a field of 10.
Here is a "window" on the outcome, taken from racenet:

Screen Shot 2021-03-18 at 6.14.32 pm.png
https://www.racenet.com.au/horse-racing-results/rosehill-20210313
.
I would suggest that the pattern this instance illustrates is how "punters" view their gambling. This
is "one amateur observer's assessment" of the psychology of gambling. It is intended as a gentle
description of an area of behaviour that attempts to see objectively how it plays out.

In a nutshell, typical "punters" focus on their wins, and minimise their attention to the losses. This may
well be how they talk to themselves about their performance as well as how they talk about it to their friends.
It's great to report on a win, the bigger the better, especially if it had been announced as a prediction before
the event. But a loss is minimised, by not being discussed, or by being offset by "circumstances" (the horse
"placed", or was winning until the last 100m, or for the punter's ten bets of the day the wins "almost offset"
the losses, etc). The main motivations for placing a bet are the pleasure and self-esteem generated by winning. The negative emotion associated with not winning is minimised, because that was not what was being sought, and focus easily returns to the positive emotions of the win, which was the objective...

Perhaps Puffin would care to comment on whether, in his experience, this is a realistic description...

/RogerE

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Re: Patterns of Prediction

Post by RogerE »

And now for something completely different... [Cue the Monty Python theme music]

Gaps in the Prime Number Sequence

Primes are the Building Blocks

The prime numbers are of fundamental interest because they are the multiplicative building blocks
for all the positive integers. The sequence begins
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...
.
The missing numbers are products of collections of primes. Because they are composed of primes, they are called composite numbers, or "composites" for short (just as prime numbers are called "primes" for short).
Every composite is the product os a unique collection of primes. A few early instances:
2x2 = 4, 2x3 = 6, 3x3 = 9, 2x2x3 = 12, 3x3x3 = 27, 2x3x5 = 30,
3x17 = 51, 2x2x2x2x2x2 = 64, 3x29 = 87, 7x13 = 91, 7x17 = 119, 11x13 = 143, ...
.
Can you decide whether 1001 is prime or composite? How about 8629, 8631 and 8633? (Hint: just one is prime.) That brings us to the next topic.

Factorising is Hard

The general problem of deciding whether a large number is prime or composite is notoriously difficult. Even if
a given number is known to be composite, finding its prime factors can still be extremely difficult.

Using this principle, there are codes which can be constructed in which a large number A is the basis for the code, and the decryption of any secret message in that code can only be done if the receiver knows the prime factors of A. It is relatively straightforward to take two very large prime numbers B and C and multiply them together to construct A = BxC. An authorised recipient is told the numbers B and C, so can decode messages which use A as their basis. An unauthorised interceptor of messages might know the number A, but in order to decrypt the messages the number A would first have to be factorised into the prime product BxC to reveal the B and C essential for decryption. For the code maker, multiplying known primes B and C together "easily" gives A, but for an eavesdropper, unscrambling the egg to take A apart and recover the prime factors B and C is vastly more difficult...

Gaps between Primes

The gaps between consecutive primes are one of the first things that attract our attention when we look at the sequence of primes and try to discern patterns in that sequence. After 2, all subsequent primes are odd numbers, so the gap between any two is even:
3 - 2 = 1, 5 - 3 = 2, 7 - 5 = 2, 11 - 7 = 4, 13 - 11 = 2, 17 - 13 = 4, ...
.
So, after the first term 1, the "first difference" sequence for the primes only has even terms:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
.
On average, the gap terms increase in size. However, it seems there is no end to the terms equal to "2" in the sequence, but no one has been able to prove or disprove that. (Each "2" corresponds to a pair of consecutive odd numbers which are both prime. These are twin primes, and the Twin Prime Conjecture predicts that such pairs occur endlessly in the sequence of primes.)

It is easy to show that the gap between consecutive primes can be larger than any chosen number. Let me give a concrete example by showing that there must be consecutive primes with difference at least 10.

First construct the number M = 2x3x5x7 = 210, made from the first four primes. Consider the nine consecutive numbers
M + 2, M + 3, M + 4, ..., M + 10.
Since M is a multiple of 2, so M + 2, M + 4, M + 6, M + 8, M + 10 must all be multiples of 2; therefore they are all composite. Since M is a multiple of 3, so M + 3, M + 6, M + 9 must be multiples of 3; therefore they are all composite. Since M is a multiple of 5, so M + 5 and M + 10 must be multiples of 5, so are composite. Since M is a multiple of 7, so is M + 7, and therefore M + 7 is composite. This shows that
M + 2, M + 3, M + 4, ..., M + 10
are all composite. Let p be the largest prime number less than M + 2, and q be the smallest prime number greater than M + 10. Then p and q are consecutive primes, with difference q - p is at least (M + 11) - (M + 1) = 10. Hence p and q are a pair of consecutive primes that differ by at least 10.

In this concrete example, M + 1 = 211 is the prime p. It turns out that M + 11 = 221 = 13x17 is composite, as is M + 12 = 222 = 2x3x37, but M + 13 = 223 is the prime q, so in this case q - p = 223 - 211 = 12. The construction only ensured that we would find a "gap" of size at least 10, but in practice we have found the consecutive primes 211 and 223, with a gap of size 12,

Now notice that if M* is the result of multiplying M by any power of 2, the same reasoning shows that M* + 2, M* + 4, M* + 6, M* + 8, M* + 10 are all multiples of 2, so are composite; M* + 3, M* + 6, M* + 9 are all multiples of 3, so composite; M* + 5, M* + 10 are multiples of 5, so composite; and M* + 7 is a multiple of 7, so composite. Hence there are no primes in the nine consecutive numbers M* + 2, M* + 3, ..., M* + 10. If p* is the largest prime less than M* + 2, and q* is the smallest prime greater than M* + 10, then q* - p* ≥ 10. Because there are infinitely many choices for M*, it follows that there are infinitely many pairs of consecutive primes with difference 10 or more.

As a few concrete examples, the construction leads us to discover that
421 and 431 are consecutive primes, with difference 10;
839 and 853 are consecutive primes, with difference 14;
1669 and 1693 are consecutive primes, with difference 24.
Specifically:
if M* = 2M = 420, then M* + 1 = p* = 421 and M* + 11 = q* = 431, so q* - p* = 10;
if M* = 4M = 840, then M* + 1 = 841 = 29x29, so M* - 1 = p* = 839; M* + 11 = 851 = 23x37,
so M* + 13 = q* = 853;
then q* - p* = 14;
if M* = 8M = 1680, then we have a remarkable patch of composites:
M* + 1 = 1681 = 41x41; M* - 1 = 1679 = 23x73;
tweaking the original construction shows that M* - 2, ..., M* - 10 will all be composites:
in particular, M* - 3 = 1677 = 3x13x43; M* - 5 = 1675 = 5x5x67; M* - 7 = 1673 = 7x23;
M* - 9 = 1671 = 3x557; so it turns out that M* - 11 = p* = 1669.
Also M* + 11 = 1691 = 19x89, so it turns out that M* + 13 = q* = 1693.
Then p* = 1669 and q* = 1693 are consecutive primes, with difference q* - p* = 24.

The construction of M can be modified by taking the product of any initial collection of primes, and in that generalised form it gives us a "constructive" proof that
For any chosen number N, there are infinitely many pairs of consecutive primes
with difference greater than N.

.
I will continue the discussion of gaps in the sequence of primes in another post.

/RogerE :D

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Re: Patterns of Prediction

Post by Ubobo.R.O. »

RogerE wrote:
18 Mar 2021 19:15
Perhaps Puffin would care to comment on whether, in his experience, this is a realistic description.../RogerE.
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Re: Patterns of Prediction

Post by RogerE »

Thanks for that brief but clear endorsement Puffin. :D

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Re: Patterns of Prediction

Post by RogerE »

Gaps in the Prime Number Sequence cont.


There is a Stampboards thread called "Counting up in prime numbers". Each post in that thread contains the next prime number after the previous post's contribution, and if a dense cluster of primes is encountered (such as two consecutive odd numbers which are both prime, or three primes among four consecutive odd numbers) then all members of the cluster are listed in the same post.

Primes around 114,660

The twin primes 114659 and 114661 drew my attention to the interesting fact that the intermediate number N = 114660 is a "pivot point" for many primes in its neighbourhood. Not only are N - 1 and N + 1 both primes, but any other neighbouring number N - q or N + q can only be prime if q itself is prime when 1 < q < 121. This is discussed/explained in the post:
https://www.stampboards.com/viewtopic.php?f=11&t=85523&p=6949208&#p6949208
.
It hinges on the fact that N has a cluster of small prime factors, N = 2x2x3x3x5x7x7x13.
If q is any composite number in the range 1 < q < 121 = 11x11, then the smallest prime factor of q is less than 11. Then q shares its smallest prime factor with N, so N - q and N + q are both multiples of that smallest prime factor, so are composite. Even if q is prime it does not follow that N - q or N + q will be prime; but what does follow is that if q is in the range 1 < q < 121, then N - q or N + q can only be prime when q itself is a prime number between 11 and 113 (inclusive), but not equal to 13. A tabulation of all the possibilities is given in the designated post.

Incidentally, this turns up the interesting pair of consecutive primes N + 53 = 114713 and N + 83 = 114743, an instance of a large gap, of size 30.

Primes between 6300 and 6600

I recently made a sequence of posts based on the primes between 6300 and 6600. My attention was caught by the rather large gaps 24, 22,18, 30 and 18 which all occur between consecutive primes in the interval from 6397 to 6547. This motivated me to study whether the odd number M = 6435 can be regarded as a "pivot point" for many primes in its neighbourhood.

M = 6435 = 3x3x5x11x13 is a nice collection of small prime factors with 7 and 17 being the first two odd primes that are absent. The difference between M and some odd number n in its neighbourhood is an even number, say 2Q, since M and n are both odd. Let q be the largest odd divisor of Q (so q results from Q by dividing out all factors 2 present in Q). For instance, if n = 6379 then 2Q = M - n = 6435 - 6379 = 56. Then Q = 28, and q = 7.

In order for n to be prime, it is necessary that q have no factor in common with M, so the prime factors of q can only be 7 or primes 17 or higher. The two smallest composite values possible for q are 7x7 = 49 and 7x17 = 119.

Summary: If q is less than 119, the only possibilities for n to be prime are q = 1, q = 7, q = 49 or q is a prime between 17 and 113 (inclusive).

This can be turned around to give a practical prediction about which numbers n falling strictly between M - 2x119 = 6197 and M + 2x119 = 6673 can be prime. They are the numbers of the form n = M – 2Q or n = M + 2Q, where 2Q is restricted to these possible values of 2Q:
q = 1 —> 2Q = 2, 4, 8, 16, 32, 64, 128
q = 7 —> 2Q = 14, 28, 56, 112, 224
q = 49 —> 2Q = 98, 196
q = 17 —> 2Q = 34, 68, 136
q = 19 —> 2Q = 38, 76, 152
q = 23 —> 2Q = 46, 92, 184
etc.
For each actual prime P between 6300 and 6600, here is a tabulation showing P, "P step" (the gap between consecutive primes), Q = ½|M – P|, and q (largest odd divisor of Q):

Screen Shot 2021-03-18 at 5.56.05 pm.png
Screen Shot 2021-03-18 at 5.57.29 pm.png

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

The Patterns of Prediction thread notes this prediction for number of views of the Happy Day thread:
Ubobo.R.O. wrote:
23 Mar 2021 11:53
I will be holding a giveaway when "total views" reaches 100,000.
.
Image
.
I calculate in 15 days time on Apr 7th. There will also be a giveaway on Apr 25th Maggie's birthday. 8-)
The Crystal Ball — Proven Prediction Device
The Crystal Ball — Proven Prediction Device
.
My recent experience with predictions about when our chatboard Stampboards would reach a particular
number of posts, or a particular number of members, were not particularly accurate. It turned out that
the approaching round number record (for the first) tended to stimulate "enthusiastic participation" by
members, so human psychology foiled the objectivity of the predictions... It will be interesting to see
how the Happy Day thread prediction fares in practice.

/RogerE :D

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Re: Patterns of Prediction

Post by Ubobo.R.O. »

I'm running with it. Find me a bookie !

2021-03-23_115916.jpg
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Re: Patterns of Prediction

Post by RogerE »

Coin tossing predictions

Here is an extract about coin tossing which I will acknowledge in a subsequent post:

Screen Shot 2021-03-24 at 2.22.04 am.png
.
Try answering this challenge before you go to my next post, where I will discuss the question
and identify/acknowledge the source.

/RogerE :D

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Re: Patterns of Prediction

Post by norvic »

bazza4338 wrote:
13 Jan 2021 08:35
RogerE wrote:
12 Jan 2021 19:40
Patterns of Prediction

Interestingly, it appears no country has yet issued a stamp commemorating Norbert Wiener, arguably the leading mathematical pioneer of the theory of stochastic processes.

ImageImageImage
An example of Common Law, or the laws against fraud and misrepresentation? The final image is not a postage stamp, being totally illegal, although interesting as it is so obscure compared with similar stamps showing Marilyn Monroe, Michael Jackson, Elvis, or Mickey Mouse.
Ian Billings - Norvic Philatelics - Remember almost everything I picture is available, unless otherwise stated or copied from elsewhere, as I reduce a roomful of 'stuff' - just ask. GB stamps info: https://blog.norphil.co.uk, NPhilatelics on twitter, www norphil.co.uk, shop.norphil.co.uk for our e-commerce site

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Re: Patterns of Prediction

Post by RogerE »

So, norvic, what about my coin tossing question?

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Coin tossing predictions cont.

I recently posted the following extract from the article
Mark F. Schilling, The longest run of heads,
College Mathematics Journal, v.21, n.3, May 1990, pp.196-207
https://www.maa.org/sites/default/files/images/upload_librar ... p0021g.pdf

Image
.
I asked readers to try answering this challenge. It's now time to share Schilling's comments about it.

Screen Shot 2021-03-29 at 2.54.29 am.png
.
Screen Shot 2021-03-29 at 2.56.05 am.png
.
Schilling follows this with a nice discussion around calculating the probability of various longest runs for n tosses. He first takes the case n = 3, and then considers the general case. (I will not try to summarise the details, but any interested reader can find them by going to the link I've provided to Schilling's paper.)

Here are Schilling's histograms showing the probability distribution for various longest run lengths in three cases; n = 50, 100, 200. (The last matches the opening discussion around 200 successive coin tosses.)

Screen Shot 2021-03-29 at 2.50.25 am.png
.
The top histogram shows that for n = 50 tosses, the most probable longest run of heads is 4, and the probability that it is 4 or 5 is over 50%.
The middle histogram shows that for n = 100 tosses, the most probable longest run of heads is 5, and the probability that it is 5 or 6 is over 50%.
The bottom histogram shows that for n = 200 tosses, the most probable longest run of heads is 6, and the probability that it is 6 or 7 is over 50%.

A striking feature of these distributions, discussed by Schilling, is that doubling the number of tosses "shifts" the probability distribution one place to the right, while scarcely changing its overall shape. (This is a "logarithmic growth" in the probability distribution as a function of n, the number of tosses.)

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Keeping an eye on a recent prediction:

Screen Shot 2021-04-04 at 2.58.41 am.png
.
Looking good...

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

The result of the Ubobo.R.O. prediction about when the number of views of the Happy Day
thread would reach K** = 100,000:

2021-04-06_200415.jpg
.
Terry felt disappointed that his prediction happened 4hrs before 7 April started, his predicted "event day".
He needn't mind — it was remarkably close, and it was probably also hastened by the human factor of Stampboarders taking an interest and checking on progress as the K** round number approached...
Recall that the prediction appeared on this thread on 23 March, when the view total was 98K:

2021-03-23_104936.jpg
.
/RogerE :D

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Re: Patterns of Prediction

Post by castores »

Hi RogerE,

Back to your original post on this thread (and many thanks for starting it).
Today I came across Marchetti's constant for the first time.

Following up leads to learn about it has caused me to think about pattern seeking more generally.
In turn this has caused me to reflect that pattern seeking is a very large topic
Pattern recognition as a human trait

It appears to be a "hard wired" human trait for us to try to organise our experiences of the real world by seeking patterns, so we can better understand and predict the outcomes of similar events.
I recently had to ring a number and it immediately looked 'nice' to me.

I recognise certain numbers will instill pictures and thoughts in the brain (1066, 1999, 999,999, 666, 888, etc).
Sometimes its nature or clouds that we see 'things' or 'patterns' in.

And sometimes the numbers just look good?

This number had one 3, one 9, it had three 4s and three 8s. I could start exaggerating about various things that occurred to me with this number, bottom line the numbers put together just looked great.

And as a human will put images together from clouds we do this with numbers as well.

Hard-wired, yes, and from there can be 'refined' if that's the correct word.

Edit

I have just seen the above post and agree, yes, the countdown probably increased viewing :)
(and I've seen screen dumps of 99,999 & 100,000 showing me as last post - sorry Ubobo.R.O. - that should have gone to you :oops: )

(I.... just.... predicted.... it..... :D )

(by the way, the 8s were 888)
Last edited by castores on 08 Apr 2021 03:43, edited 1 time in total.
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Re: Patterns of Prediction

Post by RogerE »

Hello castores. I'm glad you're finding happy ideas and insights
in this thread. Tha :) nks
/RogerE

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Re: Patterns of Prediction

Post by RogerE »

Coin tossing — further thoughts

Runs in a random sequence are patterns which easily attract our attention. If the sequence has just two
possible symbols, like H and T for coin tossing, or 0 and 1 for binary numbers, the runs are most noticeable.

It is reasonable to describe any sequence composed of just two possible symbols as a binary sequence, and any such sequence can be "converted" to a sequence of zeros and ones by simple substitution. For example, the coin tossing sequence TTHTHHT has its "structure" perfectly well represented by 1101001. This is particularly helpful if we are only interested in the patterns within the sequence, and we don't care whether the conversion was actually 1 = T, 0 = H or alternatively whether it was 1 = H, 0 = T.
.
download.jpg
.
So, let's agree that to study the structure of random binary sequences, we will represent each one as a binary number — a sequence with the possible symbols 0 and 1, and always beginning with 1 so the sequence corresponds to the "proper" binary notation for a positive number. For example, the patterns present in the sequences HHTTTHT and TTHHHTH will both be treated as the patterns in the binary number 1100010. The sequence has length 7, and consists of 4 runs. It begins with a run of length 2, followed by a run of length 3, and ends with two runs each of length 1.

So, here are some questions to consider:
• What would you guess is the most likely number of runs in a random binary sequence of length 50?
• We have recently seen a discussion of the most likely length of a longest run in a random binary sequence of length 50. What would you guess is the average run length in a random binary sequence of length 50?

And here are some actual calculations you might like to try:
How many binary numbers have length 9?
• What is the total number of runs making up all the binary numbers of length 9?
• What is the average number of runs in a binary number of length 9?
• What is the average run length in a binary number of length 9?

I will allow two days for you to consider your answers. Then I will post my "official solutions".

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Coin tossing — further thoughts (2)

In my previous post I asked readers the following challenge questions:
So, here are some questions to consider:
• What would you guess is the most likely number of runs in a random binary sequence of length 50?
• We have recently seen a discussion of the most likely length of a longest run in a random binary sequence of length 50. What would you guess is the average run length in a random binary sequence of length 50?

And here are some actual calculations you might like to try:
How many binary numbers have length 9?
• What is the total number of runs making up all the binary numbers of length 9?
• What is the average number of runs in a binary number of length 9?
• What is the average run length in a binary number of length 9?
The answers are: 25.5; approximately 1.96; 256; 1280; 5; 1.8.
Here are some details about how they were established:

Screen Shot 2021-04-14 at 10.03.24 pm.png
.
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

"Round numbers" are basically pattern numbers, and noticing them gives us pleasure.
Of course, they are artefacts of the number system we use — in our case, that's normally
the decimal system.

I have remarked elsewhere that the scale
K = 1000, K* = 10,000, and K** = 100,000
provides a nice "order of magnitude" basis for recognising decimal round numbers,
such as when it comes to celebrating particular posting milestones reached by
Stampboards members.

Intermediate round numbers are less "significant" and this goes for submultiples
of those scale markers. Nevertheless, some intermediate round numbers have
a modest appeal. For instance, this post is my post number 2¾K*.
That is sufficiently "round" to give me pleasure in noticing it ;)

Screen Shot 2021-04-21 at 1.35.32 am.png
.
Screen Shot 2021-04-21 at 1.38.50 am.png
.
So much for self-referential posting ;)

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Prediction based on close acquaintance with relevant factors (thanks to
the Happy Day thread mediated by Ubobo.R.O.)

9675c.jpg
.
Thanks Terry!

This SWAMP! cartoon reminded me of one of my all time favourites:

download.jpg
.
/RogerE :D

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