The Formulae
Below is the table of the Pick 3 Sums. It's highlighted in some key areas that will help in finding the formulae that describe it. The first highlights are the black-white to indicate the symmetry of the distribution in the Sum column. The second highlights are the red-green-blue and gray used to indicate where the distribution breaks quantum continuity; the gray highlight is special due to the overlap of formulae.
Pick 3 Sum Distribution |
Sum |
Frequency |
0 |
1 |
1 |
3 |
2 |
6 |
3 |
10 |
4 |
15 |
5 |
21 |
6 |
28 |
7 |
36 |
8 |
45 |
9 |
55 |
10 |
63 |
11 |
69 |
12 |
73 |
13 |
75 |
14 |
75 |
15 |
73 |
16 |
69 |
17 |
63 |
18 |
55 |
19 |
45 |
20 |
36 |
21 |
28 |
22 |
21 |
23 |
15 |
24 |
10 |
25 |
6 |
26 |
3 |
27 |
1 |
Looking at the sequence of the frequency for the sums 0 to 9, it appears to be a function of a special sequence known as the Arithmetic Series. The Arithmetic Series is the sum of a fixed set of numbers that change in a known pattern, typically by a function like n, 2n - 1, or n2. The summed pattern then results in a different equation. In the table, the arithmetic series is determined to be ( n (n + 1) ) / 2. However, due to the offset of the Sum being 1 less than the n value for that corresponding Frequency, the equation becomes ( (n + 1) ((n + 1) + 1) ) / 2 or ( (n + 1) (n + 2) ) / 2 by morphing n --> n + 1. This is to offset the Sum value so it can be used to determine the Frequency value in terms of n.
On the other side of the symmetry, sum 18 to 27 are the same as the 0 to 9 values only reversed. Figuring out how to express this as an equation is fairly simple; by morphing the n value as it relates to the 0 to 9 sequence. That morph is n --> 28 - n and is then plugged into the ( n (n + 1) ) / 2 equation by replacing the n values with 28 - n. The equation is then found as follows: ( (28 - n) ((28 - n) + 1) ) / --> ( (28 - n) (29 - n) ) / 2.
Now, finding the inner sequence of the sums for 10 to 17 is a bit trickier. To find it, we need to know how each quantity is derived for the whole table. This will also support the understanding of how the other sequences relate to each other. We work this by doing an Expansion of Process; starting at a lower more simple equation or method and working to a more complex equation or method.
We begin at a Pick 1 Sums level and increase the Expansion to the Pick 2 Sums and then the Pick 3 Sums. The table below shows what a Pick 1 Sums would look like and it's pretty simple.
Pick 1 Sum Distribution |
Sum |
Frequency |
0 |
1 |
1 |
1 |
2 |
1 |
3 |
1 |
4 |
1 |
5 |
1 |
6 |
1 |
7 |
1 |
8 |
1 |
9 |
1 |
The Pick 2 Sums is related to the Pick 1 Sums distribution by a simple offset progression. If the Pick 2 numbers are X Y, then the sum is X + Y. Let X be a Pick 1 sum, Y be separate value, and both be a subset of the Pick 2 set. The progressive increment is shown in the following table.
Pick 2 Sum Distribution |
Sum |
Subset Frequencies |
Frequency |
X = 0 to 9 |
Y = 0 |
Y = 1 |
Y = 2 |
Y = 3 |
Y = 4 |
Y = 5 |
Y = 6 |
Y = 7 |
Y = 8 |
Y = 9 |
Total |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
2 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
3 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
4 |
4 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
5 |
5 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
6 |
6 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
7 |
7 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
8 |
8 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
9 |
9 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
10 |
10 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
9 |
11 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
8 |
12 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
7 |
13 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
6 |
14 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
5 |
15 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
4 |
16 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
3 |
17 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
18 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
The Pick 1 distribution moves in a progressive fashion which then give rise to the total frequency of the Pick 2 distribution. This same process can be carried over to the Pick 3 Sums using the Pick 2 Sums distribution, shown below. If the Pick 3 numbers are X Y Z, then the sum is X + Y + Z. Let X Y be a Pick 2 sum, Z be separate value, and both be a subset of the Pick 3 set.
Pick 2 Sum Distribution |
Sum |
Subset Frequencies |
Frequency |
X Y = 0 0 to 9 9 |
Z = 0 |
Z = 1 |
Z = 2 |
Z = 3 |
Z = 4 |
Z = 5 |
Z = 6 |
Z = 7 |
Z = 8 |
Z = 9 |
Total |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
2 |
3 |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
6 |
3 |
4 |
3 |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
10 |
4 |
5 |
4 |
3 |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
15 |
5 |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
0 |
0 |
0 |
21 |
6 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
0 |
0 |
28 |
7 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
0 |
36 |
8 |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
45 |
9 |
10 |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
55 |
10 |
9 |
10 |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
63 |
11 |
8 |
9 |
10 |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
69 |
12 |
7 |
8 |
9 |
10 |
9 |
8 |
7 |
6 |
5 |
4 |
73 |
13 |
6 |
7 |
8 |
9 |
10 |
9 |
8 |
7 |
6 |
5 |
75 |
14 |
5 |
6 |
7 |
8 |
9 |
10 |
9 |
8 |
7 |
6 |
75 |
15 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
9 |
8 |
7 |
73 |
16 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
9 |
8 |
69 |
17 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
9 |
63 |
18 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
55 |
19 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
45 |
20 |
0 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
36 |
21 |
0 |
0 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
28 |
22 |
0 |
0 |
0 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
21 |
23 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
3 |
4 |
5 |
15 |
24 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
3 |
4 |
10 |
25 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
3 |
6 |
26 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
3 |
27 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
As you can see, the sequence the leads to the total frequency can now be determined by the progression of the sub frequency sequences. The sub frequencies sum together to form the total frequency. Looking at the green highlighted section for sums 10 to 17, the sums frequency is a unique folded or doubled arithmetic sum; where part of the summation is actually a subtraction of a smaller portion of the arithmetic sequence. Example: Sum 11 is the sum of the subset frequencies {8, 9, 10, 9, 8, 7, 6, 5, 4, 3}. That set is the subset of a common denominating set, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}, and that set is the subset of Doubled Arithmetic Sum Superset, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}.
Too make this fairly short, the Doubled Arithmetic Sum Superset is use to find the Arithmetic Sum of the 10 to 17 Sums by basically subtracting the stuff we don't need. In the case of the 11 sum, the parts that are subtracted are the individual sums of the following sets: left - {1, 2, 3, 4, 5, 6, 7}, middle - {10}, and right - {2, 1}. Each of those sets can be found mathematically and relative to the sum it is being derived from. The Double Arithmetic Sum is a constant, 2 ( ( 10 (10 + 1) ) / 2 ) --> 2 ( ( 10 (11) ) / 2 ) --> 2 (110 / 2) --> 110. The left value is ( (18 - n) ((18 - n) + 1) ) / 2, middle value is a constant 10, and right value is ( (n - 9) ((n - 9) + 1) ) / 2. Put them all together properly we get, 110 - [ ( (18 - n) ((18 - n) + 1) ) / 2 ] - [ 10 ] - [ ( (n - 9) ((n - 9) + 1) ) / 2 ].
Each of the three equations make up the whole distribution for Pick 3. In the gray highlighted frequencies, the equations are equal at those points. The following are the equations and what they reduce down to going through the math to expand and collect the n terms in each.
((n + 1)(n + 2)) / 2 |
110 - [((18 - n)((18 - n) + 1)) / 2 ] - [10] - [((n - 9)((n - 9) + 1) ) / 2] |
((28 - n)(29 - n)) / 2 |
|
|
|
Later we will replace the n value with x and set it equal to y for further quantum morphing.
Note - We'll posting this in chunks so as not to over whelm you while viewing it. Look for further postings.