Cool Lotto Frequently Asked Questions

Tell us a bit more about those Punch and Spread coefficients.

This is actually more complicated that you would suspect. The idea comes from mixing some notions taken from advanced theoretical Physics and Pure Mathematics.

In Physics there is a concept of the 'least action principle'. (That minimizes the Lagrangian of the action). It describes behavior of statistical entities by minimizing something which is obtained by subtracting kinetic and potential energies of the object. One may describe complex statistical processes in this manner (like for example the motion of the light passing through complex interfaces).

In Mathematics there is something called Banach vector spaces. These are 'vector spaces' with a norm defined in them. Banach is the name of the man who invented them in the first half of the twentieth century.

The concept behind the Punch and the Spread is to apply the principle of the least action to the Banach space defined on the random numbers selected from the playing board. The Punch and the Spread are a bit like the potential and kinetic energy in the least action principle in physics. In physics you calculate the minimum by finding the derivative and equating it to zero. It can be done like this because the spectrum considered in physical processes is of continuous nature (therefore a derivative may be calculated).

In the case of finite random number collection (like in the case of the lotto game), instead of taking the derivative I calculate the minimum using the computer (it is finite and requires a finite number of steps that may be done numerically). The program defines a norm on the finite number collection present in the game, and then builds a Lagrangian using this norm. Subsequently the computer estimates the minimum of the Lagrangian. This in turn provides the Punch and the Spread listed by the program.

In Physics the idea of minimizing the Lagrangian seems to be a bit magical. This is because the difference of the kinetic and potential energies does not have a natural physical interpretation. They sum is just the total energy of the system which is very natural to understand and visualize. On the other hand, the difference seems to be meaningless.

Nevertheless, this is the difference of the energies that gives rise to the notion of the Lagrangian and the possibility of finding the least action. In turn the minimum provides the equations of motion (you may derive the Newton laws in this manner if you like). Similarly the minimum of the Lagrangian built on the Banach space defined in here may seem to be not natural, but the effect is the estimation of the game result.

You may read about least action principle in any of these:

http://arxiv.org/ftp/arxiv/papers/0704/0704.0880.pdf

http://en.wikipedia.org/wiki/Action_(physics)

http://www.eftaylor.com/software/ActionApplets/LeastAction.html

You may read about Stefan Banach and Banach spaces in any of these:

http://en.wikipedia.org/wiki/Stefan_Banach

http://en.wikipedia.org/wiki/Banach_space

http://www.math.okstate.edu/~alspach/banach/

What values the coefficients should be to represent realistic winning combination?

Both coefficients should be at least 60 in order to represent the winning combination. In fact you should be able to verify that at least 70% of the already drawn lotto number combinations should have both Punch and Spread values exceeding 60%. Both Punch and Spread are normalized to be in the range of 0 to 100 therefore one may speak in terms of percentages when quoting them.

This is a letter I wrote to someone trying to explain the logic behind Punch and Spread coefficients.

As far as my approach is concerned, the only way I could envisage as being promising is through the least action consideration. This requires some understanding of advanced Physics and Statistics.
I can give you an example so that you would know what it is all about.

Suppose you have a large glass filled with water. You could put a long object into it (like a long spoon or a knife). Let part of it be outside and part submerged. If you look at it from above you should see that the object seems to be broken at the level of water surface. At that point the knife looks like changing direction and being closer to the surface of the water. This is an optical illusion due to the fact that the light travels with different speed in water than in the air.

The speed of light in water is less than in the air. The light coming from the tip of the submerged knife travels through the water and then through the air. It chooses a particular path to do so. One may use the principle of least action to estimate that path. In fact the light travels in such a way as to take the least time to get to your eye. In a way it seems to be a thinking process. One the one hand the light movement is governed by the laws of statistics (that is it is random), on the other hand the light seems to know its starting point and the location of your eye and chooses the path which would be the quickest when traveling.

Because of this the light would initially move more toward the surface of the water (to decrease the time in the water - as it takes more time to travel through water). Then when it gets to the surface it changes the direction so to travel toward your eye. All of that in order to take the least time while traveling towards your eye.

So what does this have to do with the lotto game?
The point is that the light which is equally probable to move in any direction chooses (if you like) to travel in such a manner as to get to you in the least possible time. In case of lotto, I presume that the number combinations which are all equally probable, choose to be drawn in such a manner as to look random on the board.

The point is that the numbers could form some distinctive patterns when drawn. For example 2, 4, 6, 8, 10, 12 would be a distinctive pattern. As far as the probability is concerned this combination is as good as any other number combination. However, I have never seen such combination to be drawn. My theory is that the numbers "want" to look randomly scattered on the board, in the same sense as the light "wants" to get to your eye in the shortest possible time.

Those coefficients I have created estimate the level of pattern ordering within any given number combination. If the numbers resemble some distinctive pattern then the coefficients would decrees. The more randomly scattered the numbers the bigger the coefficients. I use quite complex mathematical formulae to get those estimations but their meaning is still the same.