d***@paradise.net.nz
2006-05-23 04:15:07 UTC
Disregarding the 'bonus number', which is only relevant
for second and fifth division prizes, the lowest combination
in NZ Lotto is {1 2 3 4 5 6}. The highest combination is
{35 36 37 38 39 40}.
Lotto lucky number combinations can be arranged in a
sequence or lexicographical order, ranked by highest lucky
numbers.
The (combination) sequence numbers are unique numbers
1-3,838,380 in the case of Lotto 6/40. They may be
calculated from the six lucky numbers in any draw and
arranged in ascending order as follows.
* Subtract 1 from each lucky number arranged in ascending
order.
* Sum the combinatorial coefficients of the resulting
numbers with 1, 2, 3, 4, 5, 6, respectively (etc.)
* Add 1.
Thus: highest combination : 35,36,37,38,39,40.
35 - 1 = 34 [x -> M] display 34
36 - 1 = 35 [2ndF] [nCr] 2 [M+] 595
36 3 7140
etc. 37 " " 4 " 66,045
38 5 501,942
39 6 3,262,623
1 [M+] 1
[RM] 3,838,380.
Proof. 40 [2ndF] [nCr] 6 = 3,838,380. (Same, equals.)
--------------------
Lotto lucky numbers (balls 1-6) Draws #1-340 were entered
into an Acorn A4000 (UK 32 bit) computer.
The sequence numbers were calculated and then sorted into
ascending order. The successive increases were calculated
and the minimum increases printed out.
A minimum difference of 113 combinations was found between
draws #331 and #220. Later, a smaller minimum difference
of 85 combinations was found between draws #85 and #339.
The average (not minimum) difference expected was
3838380/340 = 11,289.
Table.
-----------
Draw
[extract, enter sample/ keywords into groups.google.com for more.]
cheers
don.mcdonald 23.5.06.
for second and fifth division prizes, the lowest combination
in NZ Lotto is {1 2 3 4 5 6}. The highest combination is
{35 36 37 38 39 40}.
Lotto lucky number combinations can be arranged in a
sequence or lexicographical order, ranked by highest lucky
numbers.
The (combination) sequence numbers are unique numbers
1-3,838,380 in the case of Lotto 6/40. They may be
calculated from the six lucky numbers in any draw and
arranged in ascending order as follows.
* Subtract 1 from each lucky number arranged in ascending
order.
* Sum the combinatorial coefficients of the resulting
numbers with 1, 2, 3, 4, 5, 6, respectively (etc.)
* Add 1.
Thus: highest combination : 35,36,37,38,39,40.
35 - 1 = 34 [x -> M] display 34
36 - 1 = 35 [2ndF] [nCr] 2 [M+] 595
36 3 7140
etc. 37 " " 4 " 66,045
38 5 501,942
39 6 3,262,623
1 [M+] 1
[RM] 3,838,380.
Proof. 40 [2ndF] [nCr] 6 = 3,838,380. (Same, equals.)
--------------------
Lotto lucky numbers (balls 1-6) Draws #1-340 were entered
into an Acorn A4000 (UK 32 bit) computer.
The sequence numbers were calculated and then sorted into
ascending order. The successive increases were calculated
and the minimum increases printed out.
A minimum difference of 113 combinations was found between
draws #331 and #220. Later, a smaller minimum difference
of 85 combinations was found between draws #85 and #339.
The average (not minimum) difference expected was
3838380/340 = 11,289.
Table.
-----------
Draw
[extract, enter sample/ keywords into groups.google.com for more.]
cheers
don.mcdonald 23.5.06.
"Irreducible Complexity in Pure Mathematics" by Gregory Chaitin
"How precisely do you go about picking a program at random? Well, what you
do is that every time your computer asks for the next bit of the program,
you just flip a coin. So Omega is just the probability that a machine will
eventually do something. Not a big deal! ...
This may make it sound like you can calculate Omega with arbitrarily great
accuracy, just as if it were the square root of 2 or the number pi. However
this is actually impossible, because in order to do it you'd have to solve
Turing's halting problem. In fact, in a sense the halting probability Omega
is a maximally uncomputable number."
------------------------------------------------------------------
Mathematics / Randomness everywhere / pp. 319-320
C.S. Calude and G.J. Chaitin / 22 July 1999 ...
"To understand Omega better, let's compare it with another well-known
real number, pi. The digits of pi (3.1415926...) can be computed one
by one; nonetheless, if examined locally, without being aware of their
provenance, they appear `random'. People have calculated pi out to one
billion or more digits. One of the reasons for doing this, besides
breaking the world record, is the question of whether each digit occurs
the same number of times. It looks likely, but remains unproven, that
the digits 0 through 9 each occur 10% of the time in a decimal expansion
of pi. If this turns out to be true, then pi would be called a simply
normal real number. But although pi may be random in so far as it's
'normal', it is far from random in the sense of algorithmic information
theory, because its infinity of digits can be compressed into a concise
computer program for calculating them. ...
In 1998 a first unexpected result was proved: every Omega-like real number
is an Omega7. The existence of a computably enumerable random real number
that is not an Omega became less plausible, but was not ruled out. The last
step has been brilliantly accomplished by another Berkeley mathematician,
Theodore Slaman2, who has proved that every computably enumerable random
real number is Omega-like, and hence an Omega."
-------------------------------------------------------------------
http://www.iscid.org/boards/ubb-get_topic-f-6-t-000624.html
"It depends upon how one defines randomness. A random string of bits is
normally defined as a string that can't be compressed in a lesser program.
It follows that an algorithm that puts out a random string of bits must
be greater in bits than the string itself. This is randomness by design."
----------------------------------------------------------------------
"DECIDABILITY AND UNDECIDABILITY IN THE ENUMERABLE TURING DEGREES"
by Steffen Lempp
"Two fundamental notions of mathematics are those of a computable set
and of an enumerable set. A set S is called computable (or recursive)
if there is an effective algorithm which for any input x can compute
whether x is an element of S. A set S is called is called (recursively
enumberable) if there is an effective algorithm listing all elements
of S. Clearly, these notions only make sense for computable sets. ...
A set S is Turing reducible to a set T (denoted by S is less than or
equal to _T T) if there is an oracle Turing machine computing S
( with oracle T, i.e., such that the Turing machine can query
membership information about the set T). This reducibility gives
a prepartial ordering (i.e.. a reflexive and transitive relation) on
the power set of N. We can now define two sets S and T to be Turing
equivalent if they are Turing reducible to each other. This gives an
equivalence relation on the power set of N. The equivalence class of
a set S is called its Turing degree and intuitively denotes the
"information content" of the set S while stripping away all the facts
about S inessential from a computational point of view, such as
whether a particular number is an element of S."
Regards,
Stephen
"How precisely do you go about picking a program at random? Well, what you
do is that every time your computer asks for the next bit of the program,
you just flip a coin. So Omega is just the probability that a machine will
eventually do something. Not a big deal! ...
This may make it sound like you can calculate Omega with arbitrarily great
accuracy, just as if it were the square root of 2 or the number pi. However
this is actually impossible, because in order to do it you'd have to solve
Turing's halting problem. In fact, in a sense the halting probability Omega
is a maximally uncomputable number."
------------------------------------------------------------------
Mathematics / Randomness everywhere / pp. 319-320
C.S. Calude and G.J. Chaitin / 22 July 1999 ...
"To understand Omega better, let's compare it with another well-known
real number, pi. The digits of pi (3.1415926...) can be computed one
by one; nonetheless, if examined locally, without being aware of their
provenance, they appear `random'. People have calculated pi out to one
billion or more digits. One of the reasons for doing this, besides
breaking the world record, is the question of whether each digit occurs
the same number of times. It looks likely, but remains unproven, that
the digits 0 through 9 each occur 10% of the time in a decimal expansion
of pi. If this turns out to be true, then pi would be called a simply
normal real number. But although pi may be random in so far as it's
'normal', it is far from random in the sense of algorithmic information
theory, because its infinity of digits can be compressed into a concise
computer program for calculating them. ...
In 1998 a first unexpected result was proved: every Omega-like real number
is an Omega7. The existence of a computably enumerable random real number
that is not an Omega became less plausible, but was not ruled out. The last
step has been brilliantly accomplished by another Berkeley mathematician,
Theodore Slaman2, who has proved that every computably enumerable random
real number is Omega-like, and hence an Omega."
-------------------------------------------------------------------
http://www.iscid.org/boards/ubb-get_topic-f-6-t-000624.html
"It depends upon how one defines randomness. A random string of bits is
normally defined as a string that can't be compressed in a lesser program.
It follows that an algorithm that puts out a random string of bits must
be greater in bits than the string itself. This is randomness by design."
----------------------------------------------------------------------
"DECIDABILITY AND UNDECIDABILITY IN THE ENUMERABLE TURING DEGREES"
by Steffen Lempp
"Two fundamental notions of mathematics are those of a computable set
and of an enumerable set. A set S is called computable (or recursive)
if there is an effective algorithm which for any input x can compute
whether x is an element of S. A set S is called is called (recursively
enumberable) if there is an effective algorithm listing all elements
of S. Clearly, these notions only make sense for computable sets. ...
A set S is Turing reducible to a set T (denoted by S is less than or
equal to _T T) if there is an oracle Turing machine computing S
( with oracle T, i.e., such that the Turing machine can query
membership information about the set T). This reducibility gives
a prepartial ordering (i.e.. a reflexive and transitive relation) on
the power set of N. We can now define two sets S and T to be Turing
equivalent if they are Turing reducible to each other. This gives an
equivalence relation on the power set of N. The equivalence class of
a set S is called its Turing degree and intuitively denotes the
"information content" of the set S while stripping away all the facts
about S inessential from a computational point of view, such as
whether a particular number is an element of S."
Regards,
Stephen