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The combinatorics of a permutation

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I've been thinking about another topic about the many different forms of PICK 3 & PICK 4 numbers.
Some of what was covered is related to the permutational aspect, like: singles 0123, doubles 011, etc.
After going over many of the possible variations, I began to see a combinatorial aspect of the numbers.
I found there are 4 basic forms of PICK 3 and there are 8 basic forms of PICK 4 permutations.


PICK 3 Basic Permutational Forms
A A A  A A B  A B B  A B C

PICK 4 Basic Permutational Forms                                           
A A A A   A A A B   A B B B   A A B B   A A B C   A B B C  A B C C  A B C D


At this point, I began working with each basic form to find the permutational sets for each.
Something caught my mind's eye, each basic form has a combinatorial base set grouping.
The forms have either only 1 number or 2 and 3 different numbers for PICK 3.
Also, the forms have either only 1 number or 2, 3 and 4 different numbers for PICK 4.




              Game Type  
PICK 3                           
      Combinatoric Base    1       2                 
3    
Basic Permutational Form    A A A   A A B    A B B    A B C
                                                               
      Permutational Sets  1 A A A  1 A A B  1 A B B  1 A B C
                                  2 A B A  2 B A B  2 A C B
                                   3 B A A  3 B B A  3 B A C
                                                    4 B C A
                                                     5 C A B
                                                    6 C B A


              Game Type  PICK 4                                                                                   
       Combinatoric Base    1          2                                3                                   4     
Basic Permutational Form    A A A A    A A A B    A B B B    A A B B     A A B C     A B B C     A B C C    A B C D
                                                                                                                           
      Permutational Sets  1 A A A A  1 A A A B  1 A B B B  1 A A B B   1 A A B C   1 A B B C   1 A B C C   1 A B C D
                                    2 A A B A  2 B A B B  2 A B A B   2 A A C B   2 A B C B   2 A C B C   2 A B D C
                                    3 A B A A  3 B B A B  3 A B B A   3 A B A C   3 A C B B   3 A C C B   3 A C B D
                                    4 B A A A  4 B B B A  4 B A A B   4 A B C A   4 B A B C   4 B A C C   4 A C D B
                                                          5 B A B A   5 A C A B   5 B A C B   5 B C A C   5 A D B C
                                                          6 B B A A   6 A C B A   6 B B A C   6 B C C A   6 A D C B
                                                                       7 B A A C   7 B B C A   7 C A B C   7 B A C D
                                                                     8 B A C A   8 B C A B   8 C A C B   8 B A D C
                                                                       9 B C A A   9 B C B A   9 C B A C   9 B C A D
                                                                      10 C A A B  10 C A B B  10 C B C A  10 B C D A
                                                                      11 C A B A  11 C B A B  11 C C A B  11 B D A C
                                                                      12 C B A A  12 C B B A  12 C C B A  12 B D C A
                                                                                                          13 C A B D
                                                                                                        14 C A D B
                                                                                                          15 C B A D
                                                                                                          16 C B D A
                                                                                                          17 C D A B
                                                                                                          18 C D B A
                                                                                                          19 D A B C
                                                                                                          20 D A C B
                                                                                                          21 D B A C
                                                                                                          22 D B C A
                                                                                                        23 D C A B
                                                                                                        24 D C B A


I then looked for a common combinatorial base set that could be used in both PICK 3 and PICK 4 permutational sets.


Combinatoric Base Sets             
  1     2       3          4 
   A    A B     A B C      A B C D
 1 0   1 0 1  1 0 1 2     1 0 1 2 3
 2 1   2 0 2  2 0 1 3    2 0 1 2 4
 3 2   3 0 3  3 0 1 4    3 0 1 2 5
 4 3   4 0 4  4 0 1 5    4 0 1 2 6
 5 4   5 0 5  5 0 1 6    5 0 1 2 7
 6 5   6 0 6  6 0 1 7     6 0 1 2 8
 7 6   7 0 7  7 0 1 8    7 0 1 2 9
 8 7   8 0 8  8 0 1 9     8 0 1 3 4
 9 8   9 0 9  9 0 2 3    9 0 1 3 5
10 9  10 1 2  10 0 2 4    10 0 1 3 6
      11 1 3  11 0 2 5    11 0 1 3 7
      12 1 4  12 0 2 6    12 0 1 3 8
      13 1 5  13 0 2 7    13 0 1 3 9
      14 1 6  14 0 2 8    14 0 1 4 5
      15 1 7  15 0 2 9    15 0 1 4 6
      16 1 8  16 0 3 4    16 0 1 4 7
      17 1 9  17 0 3 5    17 0 1 4 8
      18 2 3  18 0 3 6    18 0 1 4 9
      19 2 4  19 0 3 7    19 0 1 5 6
      20 2 5  20 0 3 8    20 0 1 5 7
      21 2 6  21 0 3 9    21 0 1 5 8
      22 2 7  22 0 4 5    22 0 1 5 9
      23 2 8  23 0 4 6    23 0 1 6 7
      24 2 9  24 0 4 7    24 0 1 6 8
      25 3 4  25 0 4 8    25 0 1 6 9
      26 3 5  26 0 4 9    26 0 1 7 8
      27 3 6  27 0 5 6    27 0 1 7 9
      28 3 7  28 0 5 7    28 0 1 8 9
      29 3 8  29 0 5 8    29 0 2 3 4
      30 3 9  30 0 5 9    30 0 2 3 5
      31 4 5  31 0 6 7    31 0 2 3 6
      32 4 6  32 0 6 8    32 0 2 3 7
      33 4 7  33 0 6 9    33 0 2 3 8
      34 4 8  34 0 7 8    34 0 2 3 9
      35 4 9  35 0 7 9    35 0 2 4 5
      36 5 6  36 0 8 9    36 0 2 4 6
      37 5 7  37 1 2 3    37 0 2 4 7
      38 5 8  38 1 2 4    38 0 2 4 8
      39 5 9  39 1 2 5    39 0 2 4 9
      40 6 7  40 1 2 6    40 0 2 5 6
      41 6 8  41 1 2 7    41 0 2 5 7
      42 6 9  42 1 2 8    42 0 2 5 8
      43 7 8  43 1 2 9    43 0 2 5 9
      44 7 9  44 1 3 4    44 0 2 6 7
      45 8 9  45 1 3 5    45 0 2 6 8
              46 1 3 6    46 0 2 6 9
              47 1 3 7    47 0 2 7 8
              48 1 3 8    48 0 2 7 9
              49 1 3 9    49 0 2 8 9
              50 1 4 5    50 0 3 4 5
              51 1 4 6    51 0 3 4 6
              52 1 4 7    52 0 3 4 7
              53 1 4 8    53 0 3 4 8
              54 1 4 9    54 0 3 4 9
              55 1 5 6    55 0 3 5 6
              56 1 5 7    56 0 3 5 7
              57 1 5 8    57 0 3 5 8
              58 1 5 9    58 0 3 5 9
              59 1 6 7    59 0 3 6 7
              60 1 6 8    60 0 3 6 8
              61 1 6 9    61 0 3 6 9
              62 1 7 8    62 0 3 7 8
              63 1 7 9    63 0 3 7 9
              64 1 8 9    64 0 3 8 9
              65 2 3 4    65 0 4 5 6
              66 2 3 5    66 0 4 5 7
              67 2 3 6    67 0 4 5 8
              68 2 3 7    68 0 4 5 9
              69 2 3 8    69 0 4 6 7
              70 2 3 9    70 0 4 6 8
              71 2 4 5    71 0 4 6 9
              72 2 4 6    72 0 4 7 8
              73 2 4 7    73 0 4 7 9
              74 2 4 8    74 0 4 8 9
              75 2 4 9    75 0 5 6 7
              76 2 5 6    76 0 5 6 8
              77 2 5 7    77 0 5 6 9
              78 2 5 8    78 0 5 7 8
              79 2 5 9    79 0 5 7 9
              80 2 6 7    80 0 5 8 9
              81 2 6 8    81 0 6 7 8
              82 2 6 9    82 0 6 7 9
              83 2 7 8    83 0 6 8 9
              84 2 7 9    84 0 7 8 9
              85 2 8 9    85 1 2 3 4
              86 3 4 5    86 1 2 3 5
              87 3 4 6    87 1 2 3 6
              88 3 4 7    88 1 2 3 7
              89 3 4 8    89 1 2 3 8
              90 3 4 9    90 1 2 3 9
              91 3 5 6    91 1 2 4 5
              92 3 5 7    92 1 2 4 6
              93 3 5 8    93 1 2 4 7
              94 3 5 9    94 1 2 4 8
              95 3 6 7    95 1 2 4 9
              96 3 6 8    96 1 2 5 6
              97 3 6 9    97 1 2 5 7
              98 3 7 8    98 1 2 5 8
              99 3 7 9    99 1 2 5 9
              100 3 8 9  100 1 2 6 7
              101 4 5 6  101 1 2 6 8
              102 4 5 7  102 1 2 6 9
              103 4 5 8  103 1 2 7 8
              104 4 5 9  104 1 2 7 9
              105 4 6 7  105 1 2 8 9
              106 4 6 8  106 1 3 4 5
              107 4 6 9  107 1 3 4 6
              108 4 7 8  108 1 3 4 7
              109 4 7 9  109 1 3 4 8
              110 4 8 9  110 1 3 4 9
              111 5 6 7  111 1 3 5 6
              112 5 6 8  112 1 3 5 7
              113 5 6 9  113 1 3 5 8
              114 5 7 8  114 1 3 5 9
              115 5 7 9  115 1 3 6 7
              116 5 8 9  116 1 3 6 8
              117 6 7 8  117 1 3 6 9
              118 6 7 9  118 1 3 7 8
              119 6 8 9  119 1 3 7 9
              120 7 8 9  120 1 3 8 9
                        121 1 4 5 6
                        122 1 4 5 7
                        123 1 4 5 8
                        124 1 4 5 9
                        125 1 4 6 7
                        126 1 4 6 8
                        127 1 4 6 9
                        128 1 4 7 8
                        129 1 4 7 9
                        130 1 4 8 9
                        131 1 5 6 7
                        132 1 5 6 8
                        133 1 5 6 9
                        134 1 5 7 8
                        135 1 5 7 9
                        136 1 5 8 9
                        137 1 6 7 8
                        138 1 6 7 9
                        139 1 6 8 9
                        140 1 7 8 9
                        141 2 3 4 5
                        142 2 3 4 6
                        143 2 3 4 7
                         144 2 3 4 8
                        145 2 3 4 9
                        146 2 3 5 6
                        147 2 3 5 7
                        148 2 3 5 8
                        149 2 3 5 9
                        150 2 3 6 7
                        151 2 3 6 8
                        152 2 3 6 9
                        153 2 3 7 8
                        154 2 3 7 9
                        155 2 3 8 9
                        156 2 4 5 6
                        157 2 4 5 7
                        158 2 4 5 8
                        159 2 4 5 9
                        160 2 4 6 7
                        161 2 4 6 8
                        162 2 4 6 9
                        163 2 4 7 8
                        164 2 4 7 9
                        165 2 4 8 9
                        166 2 5 6 7
                        167 2 5 6 8
                        168 2 5 6 9
                         169 2 5 7 8
                        170 2 5 7 9
                        171 2 5 8 9
                        172 2 6 7 8
                        173 2 6 7 9
                        174 2 6 8 9
                        175 2 7 8 9
                        176 3 4 5 6
                        177 3 4 5 7
                        178 3 4 5 8
                        179 3 4 5 9
                        180 3 4 6 7
                        181 3 4 6 8
                        182 3 4 6 9
                        183 3 4 7 8
                        184 3 4 7 9
                        185 3 4 8 9
                        186 3 5 6 7
                        187 3 5 6 8
                        188 3 5 6 9
                        189 3 5 7 8
                        190 3 5 7 9
                        191 3 5 8 9
                        192 3 6 7 8
                         193 3 6 7 9
                        194 3 6 8 9
                        195 3 7 8 9
                        196 4 5 6 7
                        197 4 5 6 8
                        198 4 5 6 9
                        199 4 5 7 8
                        200 4 5 7 9
                        201 4 5 8 9
                        202 4 6 7 8
                        203 4 6 7 9
                        204 4 6 8 9
                        205 4 7 8 9
                        206 5 6 7 8
                        207 5 6 7 9
                        208 5 6 8 9
                        209 5 7 8 9
                        210 6 7 8 9



Once I found each set I could use them to make a combinatorial and permutational basic form representation.
Something like an (x,y) to (u,v) mapping.

Here's an example from my state's PICK 3.

Example:
    2005-11-26  2 3 3
    transform 2 3 3 into ascending order ---> 2 3 3
    transform 2 3 3 into permutational set ---> 2 = A, 3 = B then 2 3 3 = A B B
    transform A B B into basic permutational form ---> order A B B left to right from A to B then A B B = A B B
    transform A B B into combinatorial base set ---> eliminate duplicates from A B B then A B B = A B
    transform 2 3 3 into combinatorial base set ---> eliminate duplicates from 2 3 3 then 2 3 3 = 2 3
    A B B is in the basic permutational form A B B and item 1
    A B B is in the combinatorial base set 2
    2 3 is in the combinatorial base set 2 and item 18

    2 3 3 = Combinatorial Set 2, Item 18 in the Basic Permutational Form A B B, Item 1

    2005-11-25  3 7 6
    transform 3 7 6 into ascending order ---> 3 6 7
    transform 3 7 6 into permutational set ---> 3 = A, 6 = B, 7 = C then 3 7 6 = A C B
    transform A C B into basic permutational form ---> order A C B left to right A to C then A C B = A B C
    transform A B C into combinatorial base set ---> eliminate duplicates from A B C then A B C = A B C
    transform 3 6 7 into combinatorial base set ---> eliminate duplicates from 3 6 7 then 3 6 7 = 3 6 7
    A C B is in the basic permutational form A B C and item 2
    A B C is in the combinatorial base set 3
    3 6 7 is in the combinatorial base set 3 and item 95

    3 7 6 = Combinatorial Set 3, Item 95 in the Basic Permutational Form A B C, Item 2



Here's an example from my state's PICK 4.

Example:
    2005-11-26  4 1 2 1

    transform 4 1 2 1 into ascending order --->  1 1 2 4
    transform 4 1 2 1 into permutational set ---> 1 = A, 2 = B, 4 = C then 4 1 2 1 = C A B A
    transform C A B A into basic permutational form ---> order C A B A left to right from A to C then C A B A = A A B C
    transform A A B C into combinatorial base set ---> eliminate duplicates from A A B C then A A B C = A B C
    transform 1 1 2 4 into combinatorial base set ---> eliminate duplicates from 1 1 2 4 then 1 1 2 4 = 1 2 4
    C A B A is in the basic permutational form A A B C and item 11
    A A B C is in the combinatorial base set 3
    1 2 4 is in the combinatorial base set 3 and item 38

    4 1 2 1 = Combinatorial Set 3, Item 38 in the Basic Permutational Form A A B C, Item 11

    2005-11-25  7 7 1 7
    transform 7 7 1 7 into ascending order ---> 1 7 7 7
    transform 7 7 1 7 into permutational set ---> 1 = A, 7 = B then 7 7 1 7 = B B A B
    transform B B A B into basic permutational form ---> order B B A B left to right from A to B then B B A B = A B B B
    transform A B B B into combinatorial base set ---> eliminate duplicates from A B B B then A B B B = A B
    transform 1 7 7 7 into combinatorial base set ---> eliminate duplicates from 1 7 7 7 then 1 7 7 7 = 1 7
    B B A B is in the basic permutational form A B B B and item 3
    A B B B is in the combinatorial base set 2
    1 7 is in the combinatorial base set 2 and item 15

    7 7 1 7 = Combinatorial Set 2, Item 15 in the Basic Permutational Form A B B B, Item 3



I also checked to see if this covers all the PICK 3 and PICK 4 permutations.

           N! - N Factorial, N! = N * (N - 1) * (N - 2) * ... 3 * 2 * 1 and 0! = 1

      P(N,R) - Permutation, P(N,R) = N! / (N - R)!

      C(N,R) - Combination, C(N,R) = P(N,R) / R!

    P1(N,R,r) - Permutation with 1 repeating number, P1(N,R,r) = P(N,R) / r! where 1 <= r <= R

P2(N,R,r1,r2) - Permutation with 2 repeating numbers, P2(N,R,r1,r2) = P(N,R) / (r1! * r2!) where 3 <= r1 + r2 <= R

PICK 3
Set  Form  Combinatorial * Permutational = Product
1    A A A  C(10,1) = 10    P1(3,3,3) = 1      10
2    A A B  C(10,2) = 45    P1(3,3,2) = 3      135
2    A B B  C(10,2) = 45    P1(3,3,2) = 3      135
3    A B C  C(10,3) = 120   P(3,3)    = 6      720
                                       Total  1000

PICK 4
Set  Form     Combinatorial * Permutational =  Product
1    A A A A  C(10,1) = 10    P1(4,4,4)   = 1       10
2    A A A B  C(10,2) = 45    P1(4,4,3)   = 4      180
2    A B B B  C(10,2) = 45    P1(4,4,3)   = 4      180
2    A A B B  C(10,2) = 45    P2(4,4,2,2) = 6      270
3    A A B C  C(10,3) = 120   P1(4,4,2)  = 12    1440
3    A B B C  C(10,3) = 120   P1(4,4,2)   = 12    1440
3    A B C C  C(10,3) = 120   P1(4,4,2)   = 12    1440
4    A B C D  C(10,4) = 210   P(4,4)      = 24    5040
                                          Total  10000

Just a passing thought; use it any way you'd like.

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Boy I hate the way this editor massages the data...  grrrr.

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              Game Type PICK 3                           
      Combinatoric Base    1        2                  3   
Basic Permutational Form    A A A    A A B    A B B    A B C
                                                             
      Permutational Sets  1 A A A  1 A A B  1 A B B  1 A B C
                                  2 A B A  2 B A B  2 A C B
                                  3 B A A  3 B B A  3 B A C
                                                    4 B C A
                                                    5 C A B
                                                    6 C B A

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Never Mind....  Go Here for the correctly formated page --->  Comb Perm

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After working with this a while, I've found that it may be better to work with PICK 3 and PICK 4 numbers by transforming them into Natural Numbers.

Natural Numbers are all the positive integers (1, 2, 3, 4, ...).

It is easier to work with the set mathematically because it eliminates the zero.

Working with a zero mathematically can sometimes be a disaster when figuring out a quotient that has a result of zero in the denominator.

I just have to keep in mind that a result given would have to be less by one.

I came across this when trying to figure out a way to sort PICK 3 and PICK 4 numbers in ascending order.

PICK 3 was not a problem, when the numbers are in individual columns (A,B,C), I used MIN(A,B,C) MEDIAN(A,B,C) MAX(A,B,C).

However, PICK 4 was an interesting problem to get an equation that could find the two numbers between MIN and MAX.

I'll cover this another time; I'm running late.

lottaloot's avatar - AvatarZ56

Interesting Smile

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Now that I have some time, I can explain the mathematical way to sort PICK 4 numbers.

I'll handle this in an Microsoft Excel format since 'yous' guys and gals like that so much.

First, let's look at the PICK 3 method of sorting the numbers mathematically.

The numbers must be in individual columns (A, B, C).

Here's a sample from my state's Wisconsin Lottery PICK 3, 2005-12-01 to 2005-12-15.

DATE        A  B  C

2005-12-01  9  3  0
2005-12-02  6  7  6
2005-12-03  8  1  5
2005-12-04  7  8  3
2005-12-05  9  1  9
2005-12-06  8  6  0
2005-12-07  4  9  6
2005-12-08  6  1  7
2005-12-09  1  5  3
2005-12-10  0  4  3
2005-12-11  9  2  4
2005-12-12  3  0  5
2005-12-13  8  1  8
2005-12-14  5  0  4
2005-12-15  7  7  7

To get the numbers sorted in ascending order, use MIN, MEDIAN, and MAX functions.

DATE        A B C  =MIN(A,B,C) =MEDIAN(A,B,C) =MAX(A,B,C)

2005-12-01  9 3 0       0           3             9
2005-12-02  6 7 6       6           6             7
2005-12-03  8 1 5       1           5             8
2005-12-04  7 8 3       3           7             8
2005-12-05  9 1 9       1           9             9
2005-12-06  8 6 0       0           6             8
2005-12-07  4 9 6       4           6             9
2005-12-08  6 1 7       1           6             7
2005-12-09  1 5 3       1           3             5
2005-12-10  0 4 3       0           3             4
2005-12-11  9 2 4       2           4             9
2005-12-12  3 0 5       0           3             5
2005-12-13  8 1 8       1           8             8
2005-12-14  5 0 4       0           4             5
2005-12-15  7 7 7       7           7             7


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Next, let's look at the PICK 4 method of sorting the numbers mathematically.

The numbers must be in individual columns (A, B, C, D).

Here's a sample from my state's Wisconsin Lottery PICK 4, 2005-12-01 to 2005-12-15.

DATE        A B C D

2005-12-01  7 8 1 8
2005-12-02  8 7 4 6
2005-12-03  0 7 7 2
2005-12-04  2 7 4 0
2005-12-05  0 2 2 4
2005-12-06  4 5 3 9
2005-12-07  0 8 8 6
2005-12-08  9 8 1 1
2005-12-09  8 8 9 3
2005-12-10  7 0 3 0
2005-12-11  0 1 8 3
2005-12-12  0 1 0 2
2005-12-13  3 2 1 6
2005-12-14  6 2 2 4
2005-12-15  6 5 4 3

This gets a little tricky when trying to find the two middle numbers between MIN and MAX.

However, it is possible using mathematical deduction.

I have found that dealing with PICK 4 numbers can lead to a disaster when a zero is drawn.

I'll show why this is after we go through the math.

To avoid the zero, add 1 to each number; this will transform the numbers into Natural Numbers (1, 2, 3, 4,...).

                          W     X    Y     Z
DATE        A B C D     W=A+1 X=B+1 Y=C+1 Z=D+1

2005-12-01  7 8 1 8       8     9     2     9
2005-12-02  8 7 4 6       9     8     5     7
2005-12-03  0 7 7 2       1     8     8     3
2005-12-04  2 7 4 0       3     8     5     1
2005-12-05  0 2 2 4       1     3     3     5
2005-12-06  4 5 3 9       5     6     4    10
2005-12-07  0 8 8 6       1     9     9     7
2005-12-08  9 8 1 1      10     9     2     2
2005-12-09  8 8 9 3       9     9    10     4
2005-12-10  7 0 3 0       8     1     4     1
2005-12-11  0 1 8 3       1     2     9     4
2005-12-12  0 1 0 2       1     2     1     3
2005-12-13  3 2 1 6       4     3     2     7
2005-12-14  6 2 2 4       7     3     3     5
2005-12-15  6 5 4 3       7     6     5     4


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Now, we need to find the values of MIN, MAX, SUM, and PRODUCT for the values in columns W, X, Y, and Z.

These values will be used to find the middle two numbers between MIN and MAX.

The middle numbers will be X_LOW and X_HIGH.

The sorted numbers will be in ascending order in this column format: MIN  X_LOW  X_HIGH  MAX

Here's the MIN, MAX, SUM, and PRODUCT of W, X, Y, and Z:

                      W    X    Y    Z        MIN          MAX          SUM          PRODUCT
DATE        A B C D  W=A+1 X=B+1 Y=C+1 Z=D+1 MIN(W,X,Y,Z) MAX(W,X,Y,Z) SUM(W,X,Y,Z) PRODUCT(W,X,Y,Z)
2005-12-01  7 8 1 8    8    9    2    9        2            9          28              1296
2005-12-02  8 7 4 6    9    8    5    7        5            9          29              2520
2005-12-03  0 7 7 2    1    8    8    3        1            8          20              192
2005-12-04  2 7 4 0    3    8    5    1        1            8          17              120
2005-12-05  0 2 2 4    1    3    3    5        1            5          12                45
2005-12-06  4 5 3 9    5    6    4   10        4            10        25            1200
2005-12-07  0 8 8 6    1    9    9    7        1            9          26              567
2005-12-08  9 8 1 1   10    9    2    2        2            10        23              360
2005-12-09  8 8 9 3    9    9    10   4        4            10        32            3240
2005-12-10  7 0 3 0    8    1    4    1        1            8          14                32
2005-12-11  0 1 8 3    1    2    9    4        1            9          16                72
2005-12-12  0 1 0 2    1    2    1    3        1            3          7                6
2005-12-13  3 2 1 6    4    3    2    7        2            7          16              168
2005-12-14  6 2 2 4    7    3    3    5        3            7          18              315
2005-12-15  6 5 4 3    7    6    5    4        4            7          22              840

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I have to do some errands...  I'll update very soon.

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Now, the tricky math part.

Using the values of MIN, MAX, SUM, and PRODUCT we can find the values of X_LOW and X_HIGH.

Taking the SUM of W, X, Y, and Z, if we subtract the values of MIN and MAX we are left with the sum of the two middle numbers X_LOW and X_HIGH.


        Sum of X_LOW and X_HIGH:    n = SUM - MIN - MAX


Taking the PRODUCT of W, X, Y, and Z, if we divide out the values of MIN and MAX we are left with the product of the two middle numbers X_LOW and X_HIGH.


    Product of X_LOW and X_HIGH:    m = PRODUCT / (MIN * MAX)


At this point, we still don't know the values of X_LOW and X_HIGH.

However, the values can be represented as lowercase x and y.

It does not matter which is X_LOW or X_HIGH.

We can represent the sum of X_LOW and X_HIGH as:  x + y

We can also represent the product of X_LOW and X_HIGH as:  xy


        Sum of X_LOW and X_HIGH:    n = x + y

    Product of X_LOW and X_HIGH:    m = xy


We now use these equations to solve for one of the variables, preferably x.

First, isolate y in the sum equation.



          n = x + y

      n - x = x + y - x

      n - x = y

          y = n - x


Next, substitute the value of y into the product equation.


          m = xy

          m = x( n - x )

          m = nx - xx

          m = nx - x2


You math geeks should see this is a Quadratic Equation.

Now, put the Quadratic Equation in Standard Form, ax2 + bx + c = 0.


    -( nx - x2 ) + m = nx - x2 - ( nx - x2 )

          x2 - nx + m = nx - x2 - nx + x2

          x2 - nx + m = 0


From the Quadratic Equation in Standard From, we see that a = 1, b = -n, and c = m.

Using the Quadratic Formula, we can solve for x.


                                -b ± √ b2- 4ac 
    Quadratic Formula:    x =  ---------------
                                      2a


                                -(-n) ± √ (-n)2- 4(1)(m) 
                          x =  -------------------------
                                            2(1)


                                n ± √ n2- 4m 
                          x =  -------------
                                      2



The values of x are the two roots of the Quadratic Equation, this occurs from the ± symbol.

The ± symbol can also determine the X_LOW and X_HIGH values.



                                    n - √ n2- 4m 
      The X_LOW value is:      x =  -------------
                                          2
                                    n + √ n2- 4m 
    The X_HIGH value is:      x =  -------------
                                          2



We can now substitute the values of n and m from the previous equations n = SUM - MIN - MAX and m = PRODUCT / (MIN * MAX).



                                    (SUM - MIN - MAX) - √ (SUM - MIN - MAX)2- 4(PRODUCT / (MIN * MAX)) 
      The X_LOW value is:      x =  -------------------------------------------------------------------
                                                                      2



                                    (SUM - MIN - MAX) + √ (SUM - MIN - MAX)2- 4(PRODUCT / (MIN * MAX)) 
    The X_HIGH value is:      x =  -------------------------------------------------------------------
                                                                      2
As you can see, if MIN is a zero, dividing the PRODUCT by zero leads to an error.

However, because we have added a one to the numbers, we avoid getting an error and can less the values of X_LOW and X_HIGH by one to get the PICK 4 numbers.


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For those of you who don't know or are puzzled by the symbol   √  , it's a square root symbol.

          means take the square root of.

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Here's a process view of the X_LOW and X_HIGH functions in an Excel format.

Use the following functions to sort W, X, Y, and Z into MIN, X_LOW, X_HIGH, and MAX.

  MIN: =MIN(W,X,Y,Z)

 X_LOW: =((SUM(W,X,Y,Z)-MIN(W,X,Y,Z)-MAX(W,X,Y,Z))-SQRT(((SUM(W,X,Y,Z)-MIN(W,X,Y,Z)-MAX(W,X,Y,Z))^2)-4*(PRODUCT(W,X,Y,Z)/(MIN(W,X,Y,Z)*MAX(W,X,Y,Z)))))/2

X_HIGH: =((SUM(W,X,Y,Z)-MIN(W,X,Y,Z)-MAX(W,X,Y,Z))+SQRT(((SUM(W,X,Y,Z)-MIN(W,X,Y,Z)-MAX(W,X,Y,Z))^2)-4*(PRODUCT(W,X,Y,Z)/(MIN(W,X,Y,Z)*MAX(W,X,Y,Z)))))/2

  MAX: =MAX(W,X,Y,Z)

Then, readjust the values to PICK 4 numbers by subtracting one.

 UNSORTED  UNSORTED   SORTED    SORTED   
 PICK 4  W,X,Y,Z   W,X,Y,Z    PICK 4   
                    
      W=A+1X=B+1Y=C+1Z=D+1 w=MINx=X_LOWy=X_HIGHz=MAX a=w-1b=x-1c=y-1d=z-1
                    
DATEABCD WXYZ wxyz abcd
2005-12-017818 8929 2899 1788
2005-12-028746 9857 5789 4678
2005-12-030772 1883 1388 0277
2005-12-042740 3851 1358 0247
2005-12-050224 1335 1335 0224
2005-12-064539 56410 45610 3459
2005-12-070886 1997 1799 0688
2005-12-089811 10922 22910 1189
2005-12-098893 99104 49910 3889
2005-12-107030 8141 1148 0037
2005-12-110183 1294 1249 0138
2005-12-120102 1213 1123 0012
2005-12-133216 4327 2347 1236
2005-12-146224 7335 3357 2246
2005-12-156543 7654 4567 3456


This is just a wide view of the process to see what is going on.

I'll post the final unified functions of X_LOW and X_HIGH later today or tomorrow.

It will allow you to sort without having to subtract one and puting the value in another column, just one step.

Getting a little sleepy working on this lottery stuff.


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OK, here we go now.

Here's the final simplified, unified, grandiose, latest, and greatest equations/functions you'll need for sorting PICK 4 numbers into ascending order.

We'll deal with the functions as they relate to the table listed here, then show the Excel method as it relates to your spreadsheet so you can hopefully copy and paste the function into your spreadsheet.

The functions for sorting are listed in ascending order of W, X, Y, and Z.

W: =MIN(A,B,C,D)

X: =(((SUM(A+1,B+1,C+1,D+1) - MIN(A+1,B+1,C+1,D+1) - MAX(A+1,B+1,C+1,D+1)) - SQRT(((SUM(A+1,B+1,C+1,D+1) - MIN(A+1,B+1,C+1,D+1) - MAX(A+1,B+1,C+1,D+1))^2) - 4*(PRODUCT(A+1,B+1,C+1,D+1)/(MIN(A+1,B+1,C+1,D+1)*MAX(A+1,B+1,C+1,D+1)))))/2) - 1

Y: =(((SUM(A+1,B+1,C+1,D+1) - MIN(A+1,B+1,C+1,D+1) - MAX(A+1,B+1,C+1,D+1)) + SQRT(((SUM(A+1,B+1,C+1,D+1) - MIN(A+1,B+1,C+1,D+1) - MAX(A+1,B+1,C+1,D+1))^2) - 4*(PRODUCT(A+1,B+1,C+1,D+1)/(MIN(A+1,B+1,C+1,D+1)*MAX(A+1,B+1,C+1,D+1)))))/2) - 1

Z: =MAX(A,B,C,D)

The W and Z functions do not need to be modified for Natural Numbers.

They can deal directly with the PICK 4 data because there is no division involved.

 UNSORTED SORTED 
          
DATEABCD WXYZ
2005-12-017818 1788
2005-12-028746 4678
2005-12-030772 0277
2005-12-042740 0247
2005-12-050224 0224
2005-12-064539 3459
2005-12-070886 0688
2005-12-089811 1189
2005-12-098893 3889
2005-12-107030 0037
2005-12-110183 0138
2005-12-120102 0012
2005-12-133216 1236
2005-12-146224 2246
2005-12-156543 3456



Next, we can see how this looks on my system.

Here is my spreadsheet as it appears on my computer.

The PICK 4 data begins at cells B4, C4, D4, and E4 and goes to cells B18, C18, D18, and E18.

The sorted data starts at cells G4, H4, I4, and J4 and goes to cells G18, H18, I18, and J18.

If your spreadsheet is formatted differently, you'll need to modify the functions a bit to accommodate your placement of the PICK 4 data.

 ABCDEFGHIJ
1 UNSORTED SORTED 
2          
3DATEABCD WXYZ
42005-12-017818 1788
52005-12-028746 4678
62005-12-030772 0277
72005-12-042740 0247
82005-12-050224 0224
92005-12-064539 3459
102005-12-070886 0688
112005-12-089811 1189
122005-12-098893 3889
132005-12-107030 0037
142005-12-110183 0138
152005-12-120102 0012
162005-12-133216 1236
172005-12-146224 2246
182005-12-156543 3456


You can copy and paste these functions into your spreadsheet if it is formatted like this one.


Copy/Paste this MIN function to G4

=MIN(B4,C4,D4,E4)


Copy/Paste this X_LOW function to H4

=(((SUM(B4+1,C4+1,D4+1,E4+1) - MIN(B4+1,C4+1,D4+1,E4+1) - MAX(B4+1,C4+1,D4+1,E4+1)) - SQRT(((SUM(B4+1,C4+1,D4+1,E4+1) - MIN(B4+1,C4+1,D4+1,E4+1) - MAX(B4+1,C4+1,D4+1,E4+1))^2) - 4*(PRODUCT(B4+1,C4+1,D4+1,E4+1)/(MIN(B4+1,C4+1,D4+1,E4+1)*MAX(B4+1,C4+1,D4+1,E4+1)))))/2) - 1


Copy/Paste this X_HIGH function to I4

=(((SUM(B4+1,C4+1,D4+1,E4+1) - MIN(B4+1,C4+1,D4+1,E4+1) - MAX(B4+1,C4+1,D4+1,E4+1)) + SQRT(((SUM(B4+1,C4+1,D4+1,E4+1) - MIN(B4+1,C4+1,D4+1,E4+1) - MAX(B4+1,C4+1,D4+1,E4+1))^2) - 4*(PRODUCT(B4+1,C4+1,D4+1,E4+1)/(MIN(B4+1,C4+1,D4+1,E4+1)*MAX(B4+1,C4+1,D4+1,E4+1)))))/2) - 1


Copy/Paste this MAX function to J4

=MAX(B4,C4,D4,E4)



Now, select row 4, column G through J.

Copy the the selection.

Move to the next row where the additional sort data is to be placed, row 5 and column G.

Now, select row 5, column G to row 18, column G...  paste and you're done.

You can select multiple rows and columns by clicking on the cell you'd like to start with and holding the button down, then moving the pointer to whatever other cell you are trying to fill or copy.

Or, select a cell with the pointer, then press and hold the shift key, use the arrow keys to move to the next cell you are going to fill or copy.

If your spreadsheet is not like this one, you may need to modify the X_LOW and X_HIGH functions with WordPad or something to make it easier getting it into your spreadsheet.

The MIN and MAX functions are pretty easy to modify.

To modify the X_LOW and X_HIGH functions, copy and paste each into WordPad on separate lines.

Next, select the text 
B4+1,C4+1,D4+1,E4+1  and menu click Edit, then Replace.

This should come up with a dialog box with the text already in the  Find what:  text box.

Modify the numbered row and the lettered column by typing in the  Replace with:  text box.

I should like similar to this, if your PICK 4 data let's say begins at the cells F12, G12, H12, and I12, then you should type the following into the  Replace with:  text box  F12+1,G12+1,H12+1,I12+1  and click Replace All.


Now you are ready to select each function, copy, and paste it into your spreadsheet.

This was an interesting problem to solve.

I hope this helps you and gives you that much needed win.

Thank you.

End.

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