How It's Made - JADE Pick 3 Pick 4 Selector

Ok, now that things have settled down a bit. We can tell you 'how it's made'; 'it' being the number selections in JADE Pick 3 Pick 4 Selector.

The basic idea for this program came from a current Lottery Post member, Lucky. Lucky's Pick 3 and Pick 4 Followers were a great inspiration for this number selection process. Lucky's basic idea of what comes next when a particular Pick 3 or Pick 4 number combination is drawn could be applied to a basic mathematical equation related to matrix operations. That equation is called a Determinant. The Determinant allowed Lucky's idea of the followers to become more dynamic in that it now gives a direct mathematical relationship between what just happen before and what will happen next. Once again, Thank you Lucky!

Now, on to the good stuff. What is a Determinant and how is it used? A Determinant is a mathematical equation related to a square matrix that results in a single number. A square matrix is a set of numbers or variables arranged in a square pattern; like the ones listed next.

(note: some web browsers may not display the square brackets correctly. they should look like [ ].)

or

The square brackets indicate it's a square matrix. A square matrix is a special matrix that has the same number of columns as rows. The matrix can also be of any size 1 x 1, 2 x 2, 3 x 3, 4 x 4 or n x n; where n is the number of columns or rows. Below is an example of a 4 x 4 matrix.

A Determinant is a mathematical equation that takes each item in the square matrix and gives a single result. For our purposes, that's all we need to know. There are other reasons for the equation, but for what the Determinant will be doing, this will do. The Determinants we'll be looking at are, the 3 x 3 matrix and the 4 x 4 matrix. Obviously, each of these relate to their Pick 3 and Pick 4 lotteries by the number of columns or rows, 3 and 4. As an example, if the square matrix of a 3 x 3 matrix is the following.

Then the Determinant is the following mathematical equation.

(' × ' means to multiply.)

D_{3} = a × (e × i - f × h) - b × (d × i - f × g) + c × (d × h - e × g) |

The variable D_{3} is the Determinant for the 3 x 3 square matrix.

Also, if the square matrix of a 4 x 4 matrix is the following.

Then the Determinant is the following mathematical equation.

( l 'el' and o 'oh' are purposely left out so as not to confuse them with 1 'one' and 0 'zero'.)

D_{4} = | a × (f × (k × r - m × q) - g × (m × p - j × r) + h × (j × q - k × p)) - b × (g × (m × n - i × r) - h × (i × q - k × n) + e × (k × r - m × q)) + c × (h × (i × p - j × n) - e × (j × r - m × p) + f × (m × n - i × r)) - d × (e × (j × q - k × p) - f × (k × n - i × q) + g × (i × p - j × n)) |

Next, we'll look at the application of the Determinant on a 3 x 3 square matrix.

Using the follow square matrix as a guide.

The Determinant is then found using the equation, D_{3} = a × (e × i - f × h) - b × (d × i - f × g) + c × (d × h - e × g) .

If the numbers in an example matrix is as follows...

...then we can substitute the letter variables with our numbers and find the Determinant.

é | a | b | c | ù | | é | 1 | 0 | 3 | ù |

ê | d | e | f | ú | = | ê | 4 | 5 | 5 | ú |

ë | g | h | i | û | | ë | 7 | 3 | 2 | û |

Since these two square matrices are equal, we can assign each number to the variable.

a = 1, b = 0, c = 3, d = 4, e = 5, f = 5, g = 7, h = 3 and i = 2.

Now, using the equation, D_{3} = a × (e × i - f × h) - b × (d × i - f × g) + c × (d × h - e × g), we can put in the numbers.

D_{3} = 1 × (5 × 2 - 5 × 3) - 0 × (4 × 2 - 5 × 7) + 3 × (4 × 3 - 5 × 7)

D_{3} = 1 × (10 - 15) - 0 × (8 - 35) + 3 × (12 - 35)

D_{3} = 1 × (-5) - 0 × (-27) + 3 × (-23)

D_{3} = (-5) - 0 + (-69)

D_{3} = -74

The Determinant for this example is -74.

We can apply this to a list of Pick 3 numbers and find the Determinants in a drawing list. Let's say the following is a short list of Pick 3 numbers as an example.

-5 | | 1 | 0 | 3 | | | |

-4 | | 4 | 5 | 5 | | | |

-3 | | 7 | 3 | 2 | | | |

-2 | | 8 | 6 | 6 | | | |

-1 | | 1 | 3 | 7 | | | |

0 | | 2 | 6 | 1 | | | |

The oldest draw is at the top and the newest draw is at the bottom. The numbers on the far left are reference index numbers to show draw order. We can apply the Determinant equation to the list starting at the top oldest and move down to the most recent draws. The following is a step though the process for this example.

First, start at the top and find the Determinant for the first 3 draws.

-5 | é | 1 | 0 | 3 | ù | | |

-4 | ê | 4 | 5 | 5 | ú | | |

-3 | ë | 7 | 3 | 2 | û | = | -74 |

-2 | | 8 | 6 | 6 | | | |

-1 | | 1 | 3 | 7 | | | |

0 | | 2 | 6 | 1 | | | |

Next, move down one draw and find the Determinant.

-5 | | 1 | 0 | 3 | | | |

-4 | é | 4 | 5 | 5 | ù | | |

-3 | ê | 7 | 3 | 2 | ú | | -74 |

-2 | ë | 8 | 6 | 6 | û | = | -16 |

-1 | | 1 | 3 | 7 | | | |

0 | | 2 | 6 | 1 | | | |

Keep moving down.

-5 | | 1 | 0 | 3 | | | |

-4 | | 4 | 5 | 5 | | | |

-3 | é | 7 | 3 | 2 | ù | | -74 |

-2 | ê | 8 | 6 | 6 | ú | | -16 |

-1 | ë | 1 | 3 | 7 | û | = | 54 |

0 | | 2 | 6 | 1 | | | |

One more and we're done.

-5 | | 1 | 0 | 3 | | | |

-4 | | 4 | 5 | 5 | | | |

-3 | | 7 | 3 | 2 | | | -74 |

-2 | é | 8 | 6 | 6 | ù | | -16 |

-1 | ê | 1 | 3 | 7 | ú | | 54 |

0 | ë | 2 | 6 | 1 | û | = | -234 |

The numbers on the far right are the Determinants of this draw listing.

-5 | | 1 | 0 | 3 | | | |

-4 | | 4 | 5 | 5 | | | |

-3 | | 7 | 3 | 2 | | | -74 |

-2 | | 8 | 6 | 6 | | | -16 |

-1 | | 1 | 3 | 7 | | | 54 |

0 | | 2 | 6 | 1 | | | -234 |

As you can see, the Determinants can be both positive or negative and in some cases, a neutral zero. This works great as an example. However, when working with Pick 3 and Pick 4 numbers there is a number that can create a problem when looking at all the possible ways that a Determinant can generated for either Pick 3 or Pick 4. The number that can cause a problem is 0, zero. In our research, we have found that using zero can create a bias toward a Determinant value of 0. For this reason, it is best to add 1 to each Pick 3 or Pick 4 number so they are transformed from 0 through 9 to 1 through 10. This will create a more useful Determinant value to work with.

It turns out that a Determinant value of 0 is the most frequent value obtained from working the equation. Even with the individual numbers transformed from 0-9 to 1-10, the resulting Determinant with the greatest occurrence is 0 and every other value decreases in frequency from there for both positive and negative Determinants. Also, the frequency is symmetrical for positive and negative numbers. This means a positive Determinant value will have the same frequency as it's negative valued counterpart. As an example, the frequency of a Determinant 1 will have the same frequency as the Determinant -1.

We have already done the all the possible Determinant outcomes for every possible three set combination. There is in fact a total of 1 Billion possible ways to plug in a 3 x 3 matrix with ten different numbers; either 0-9 or 1-10. The frequency table for Pick 3 Determinants based on a 0-9 to 1-10 transformation set can be downloaded as Text or as an Excel sheet here, Text - Pick 3 Matrix Counts or Excel - Pick 3 Matrix Counts.

So, how does this work for picking numbers. Well, if we use the previous listing, we can see that when we advance the matrix one more draw down, we now have two of the previous draws and the unknown future picks. Example listing below.

First we need to add 1 to our Pick 3 numbers so the example has no zeros in it.

-5 | | 1+1 | 0+1 | 3+1 | |

-4 | | 4+1 | 5+1 | 5+1 | |

-3 | | 7+1 | 3+1 | 2+1 | |

-2 | | 8+1 | 6+1 | 6+1 | |

-1 | | 1+1 | 3+1 | 7+1 | |

0 | | 2+1 | 6+1 | 1+1 | |

The example draw listing now becomes the following.

-5 | | 2 | 1 | 4 | | | |

-4 | | 5 | 4 | 4 | | | |

-3 | | 8 | 4 | 3 | | | -39 |

-2 | | 9 | 7 | 7 | | | -1 |

-1 | é | 2 | 4 | 8 | ù | | 58 |

0 | ê | 3 | 7 | 2 | ú | | -278 |

1 | ë | x | y | z | û | = | D_{3} |

At this point, you can get a little glimpse of what is going to happen next. Since part of the Determinant equation is already known by the previous draws, 0 and -1, all that is needed is a way to find some reasonable values for x, y, z and D_{3}. The first thing that can be done is to plug in all the possible values for x, y and z to make a list of all the possible Determinants that can occur for the last two draws. There are only 1,000 x y z combinations to deal with for this example. Next, we can now use the Pick 3 Matrix Counts to select the best set of x y z combinations that are much more likely to be picked based on the Pick 3 Matrix Counts table. The more frequent the Determinant is the more likely that x y z combination is likely to hit.

When you look at the Pick 3 Matrix Counts table, you'll see that the highest frequency Determinants occur near 0. Also, if you were to graph the table it would look like the following.

So, now we need a way to select the best Determinant values. There are two ways to do this: 1 - Use a Classical method of an equal range above and below the 0 Determinant, 2 - Use a Quantum method based on a previous topic we posted a while back called, Random Number Transforms - Normal Distribution.

Time to let you munch on this a bit... will continue in a bit...

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Douglas Paul Smallish

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