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How far away is a P3 combo from a triple?Prev TopicNext Topic
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The One Over None
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Using the simple setup we can map all the doubles for the 2D; they are: 00, 11, 22, 33, 44, 55, 66, 77, 88, 99; where AB is the format.
It would appear as:
We can add a selection, 28 or (2, 8), to the graph to show how the selection is in relation to the doubles.
It shows the most likely closest double is 55 or (5, 5).
The doubles create a diagonal line.
If we include the cyclical numbers to show the wrap around, we can map the doubles and see where they wrap around.
As we can see 55 or (5, 5) is not the closest double to 28 or (2, 8).
The closest double is actually 00 or (0, 0)
The One Over None
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For this 2D / doubles example, it comes down to finding the shortest distance from a selection to a doubles line.
The basic equation for finding the shortest distance from a selection (A, B) to a double line is:
But, that equation only relates to the original doubles.
There are two other equations that find the shortest distance to the other cyclical doubles; they are:
and
The One Over None
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Thank you @ JADELottery
088, 389, 022, 789, 444, 555, 888.
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We can map the individual distances.
We can combined these by finding the lowest value for each position and map those lowest values.
Those values are not equally distributed.
Some have a count of 20 and some only a count of 10.
To equalize the counts, we can give those distances below a diagonal line a - (negative) value.
Below is an expanded map that includes the wrapped numbers.
In this format, there are an equal count of -5's to 4's.
The reduced map becomes the following.
It looks a lot like the meter made moves for the clockwise/counter clockwise formula.
The One Over None
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We can separate the pairs by their distance and see which pairs are closest to a distance of 0; 0 being a double itself.
The closest pairs to the doubles are d = {-1, 1) and the farthest is d = {-5}; which is essentially completely opposite of a double.
The farthest is like the two opposite sides of a clock, ie. 12 O'clock and 6 O'clock.
The One Over None
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This simple grouping of pairs works well with 2D.
When we move up in to the 3D, ABC or (A, B, C), we're going to need a way of separating the combos into manageable segments.
Like adding the - (negative) for pairs that were below a doubles diagonal, we'll need a formula for determining relative position in a 3D space.
Also, the distances in 3D don't fall into nice round numbers like {0, 1, 2, 3, …}.
We'll have to handle the real equation for distance a little differently.
The basic equation for finding a combo (point) in 3D space to a triple (line) in 3D space, where (A, B, C) is the combo (point), is as follows:
To help separate the combos further, we will be using a set of formulae we posted a while ago.
It's actually related to other posts: JADELottery's Pick 3 H-Trac System and Mapping PICK 3 data into RGB color cube.
The formulae are as follow:
Just keep these in mind, the end result we're looking for is the H value.
It is the Hue representation of the of the color mapped pick 3 combo.
The One Over None
I Know... -
We can separate the pairs by their distance and see which pairs are closest to a distance of 0; 0 being a double itself.
The closest pairs to the doubles are d = {-1, 1) and the farthest is d = {-5}; which is essentially completely opposite of a double.
The farthest is like the two opposite sides of a clock, ie. 12 O'clock and 6 O'clock.
Win!
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The 2D closest doubles are on one of three lines.
Below is a graph of the lines: two cyclical and one original double lines.
The light blue circles are the two cyclical alternate choices.
In 3D there are more alternate choices for a nearest triple.
The following graphs show the alternate choices around the original choice.
As you can see, there are 6 alternate triples that can be the closest.
The One Over None
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The original distance equation is
The 6 alternate distances equations are
These are used to initially sort the combos that are closest to a triple.
They work fine, but we end up with decimal values from the square root.
It's actually easier to just deal with the expression inside the square root to get integer values.
The equations then become the following.
The One Over None
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Using the previous equations and finding the minimum value for every possible combo, the minimum distance to a triple can be established.
Here's a table of the first set of 20 combos.
The values for d are not continuous.
We can see the distribution of d values that have a count greater than 0.
The table below shows all the possible d values and frequency count for each.
An index can be added to reference the distances in a continuous order of magnitude.
The One Over None
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The One Over None
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To further subdivide the distance values, we need to use the following formulae and refine the final H equation.
The following formulae generate Cylindrical Hue Values for numbers in a Cubic Cartesian 3D space.
The final equation needs to be tweaked a bit to keep the H value in a range of 0 to 5; a total of 6 different color ranges.
The new H equation is:
The One Over None
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The One Over None
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The One Over None
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