New Zealand Lotto

Towards a Theoretical Approach for Choosing Winning Lotto numbers.

The problem of choosing numbers without duplicates is a non-trivial one. One needs a systematic method of selecting these numbers without encountering an excessive number of duplicates. To achieve the case of no duplicates at all one can easily exhaust the available combinations. These are listed below.
	1  2  3  4  5  6
	7  8  9  10 11 12
	13 14 15 16 17 18
	19 20 21 22 23 24 
	25 26 27 28 29 30
	31 32 33 34 35 36
	37 38 39 40

Consequently we have the combination 1-6, 7-12, 13-18, 19-24, 25-30, 31-36. These would make up six tickets. Then there are four numbers left over which are 37,38,39,40. Don't worry that these are not the numbers you would choose for yourself. You could rearrange the numbers 1-40 in any order. Again its the combinations that are important and not the individual numbers. The main point to grasp here is that there are only six combinations of numbers that have no duplicates at all.

So now we have to consider the case of what happens when one number only is allowed to duplicate across selections. In the case of targetting a Division Four win duplications of one number are okay. That is if two selections have one number in common this doesn't affect your chances of winning Division Four. It is not possible to encounter multiple wins in Division Four if only one number is duplicated. (It does however affect Division Five but only if the number duplicated is the bonus number. This will be addressed at a later time.)

Consider the following example where we have 3 lotto tickets with the following number selections.

4  6  12  13  14  23
5  8  15  16  19  23
5  7  12  31  33  39

The first lotto tickets has the number 23 in common with second lotto ticket and 12 in common with the third. The second lotto ticket also has 5 in common with the third. This is illustrated in figure 1 below. For the sake of terminology I have called the instance where only 1 number is in common a 1-duplicate. Where two numbers are in common I call it a 2-duplicate. There are only 1-duplicates in the illustration below. 2-duplicates would occur if there are two numbers in common just between two lotto tickets. A 1-duplicate is less damaging to your chances of winning lotto than a 2-duplicate. The fundamental problem of lotto winning is therefore to maximise lotto tickets selections but reduce the severity of the duplicates. No duplicates are best of all, yet 1-duplicates are better than 2-duplicates.

Figure 1

For the sake of this form of analysis it makes sense only to consider those numbers that each lotto ticket has in common with another. Therefore a simpler representation can be used where the lines represent those numbers which are in common with a lotto ticket whereas the dots represent individual lotto tickets.
Figure 2

Now instead of working out which numbers are going to go on each lotto ticket it allows us the possibility to work out which numbers are going to occur as duplicates first. That is assign number to your edges first and let the nodes (tickets) take care of themselves. This leads us to a new approach to lotto number pickery. (Hmmm. Is "pickery" a word?) Say for instance I decide to choose numbers for 7 lotto tickets. I draw 7 dots on a piece of paper and draw lines from one of them.

Figure 3

Then using the second node I draw in another set of lines and carry on numbering them. This is shown in red on my diagram (figure 4) below. Notice that two of the lotto numbers have also already been chosen for the 5 remaining lotto tickets. Therefore each lotto ticket's numbers are derived from the numbers on all the lines adjoining them.
Figure 4

Now we know each node represents a lotto ticket and each edge one of the lotto numbers. Number the edges (not the nodes) and then each node's numbers can be computed from its incident edges. In this way your lotto ticket numbers can be calculated. The above scenario has been completed in the example below.
Figure 5

Now even this is not fully optimised results. We have not taken into account our original number 1-6, 7-12, etc. Remember I said that no duplicates are better than 1-duplicates. These no duplicates should be included as well. First of all I should organise it for the numbers 1-36 and leave out 37-40 initially. (It makes the explanation easier). Consider on 6 different lotto tickets the following (rather contrived) numbers. I shall represent these non duplicates as open circles.

	1  2  3  4  5  6
	7  8  9  10 11 12
	13 14 15 16 17 18
	19 20 21 22 23 24 
	25 26 27 28 29 30
	31 32 33 34 35 36

Now each of these selections do not have any numbers in common with each other. But any new selection will. (We are of course ignoring numbers 37-40 at this stage). So open circles can have corresponding lines drawn to black circles (additional lotto tickets) but not other (open) circles. Black circles can have lines drawn to open circles and also other black circles. Lines from black circles to black circles represent our 1-duplicates and lines from open circles to black represent no duplicates. Numbering the lines is left as an exercise. :)

Figure 6

I still haven't yet touched on the implications of the bonus number. Also what about the numbers 37,38,39,40.

Now this can get really hairy with pencil and paper with a decent number of lotto tickets. With $25 worth of lotto tickets you already have 50 ticket selections and that 50 nodes with 150 edges. Edges increase at three times the rate as nodes. Consequently I'm looking at writing a computer program to do all this. Those of you who are programmers should be able to appreciate that this method doesn't require you to keep track of a large number of sets and subsets of numbers. A brute force approach would require you to do this and remove duplicates by checking lotto numbers as you go and this for every number in every set. When you consider that there are three million three hundred and eight thousand, three hundred and eight different possible lotto tickets you can appreciate the computational effort involved.

Stay tuned....More to come ....

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