Appreciating the beauty of combinations

February 26, 2002 6:09 AM
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Part of the beauty of Keno is the way that numbers, or groups of numbers, combine to make a seemingly infinite possibility of deferent tickets. The beauty is not just in the variety, but also in the structure that these combinations take. There are Keno players (and writers too!) who spend years involved in the game without noticing the underlying patterns. Take the simple way ticket, grouped 2-2-2, a one way six, a three way four and a three way two spot. The ticket has a total of seven ways on it. But why?

Well, if we take a group off the ticket, we are left with 2-2, a simple one way four and two deuces, a total of three ways. And if we go one step further, we are left with a simple 2, a one way two. Thus the sequence of total ways we’ve produced is 7-3-1, by successively removing groups one by one.

Or if we start with an equally simple nine spot way ticket, 3-3-3, we also have seven total ways, a 1 way nine, a 3 way six, and a 3 way three. If we remove one group of three we have 3-3, a three way ticket: 1 way six and 2 way 3. And naturally, a group of three by itself is a one way 3. We have reproduced the 7-3-1 sequence again, this time using groups of three.

Our experiment is reproducible using even a mixed group ticket: Take nine spots grouped 4-3-2. Here we have a one way nine, a one way seven, a one way six, a one way five, a one way four, a one way three, and a one way two, for a total of seven ways! I’ll leave it to the reader to take off groups one by one, thereby producing the 7-3-1 sequence of total ways.

I will postulate here that ANY Keno ticket with three groups on it produces a total of seven ways, and consequently any subtraction of groups as above will produce in succession total ways of three and one.

What happens if we ADD a group to a way ticket?  If we add a group to our original 2-2-2 ticket with 7 total ways on it, we end up with a ticket that has 15 total ways:

2-2-2-1:  1 way 7, 1 way 6, 3 way 5, 3 way 4, 3 way 3, 3 way 3, 1 way 1.

2-2-2-2:  1 way 8, 4 way 6, 6 way 4, 4 way 2.

2-2-2-3:  1 way 9, 3 way 7, 1 way 6, 3 way 5, 3 way 4, 1 way 3, 3 way 2.

2-2-2-4:  1 way 10, 3 way 8, 4 way 6, 4 way 4, 3 way 2.

So I’ll postulate here that if you add a fourth group to a ticket that has three groups on it you’ll  produce a ticket with 15 total ways on it.  Once again I leave it to the reader to test.

So a rough rule of thumb:  When you subtract a group from a way ticket you end up with (roughly) half the total ways you had before, and when you add a group to a way ticket you end up with (roughly) double the total ways you had before. 

Or for those of you with a bent for algebra, the total number of ways on a Keno ticket is:

2^n - 1, where n is the number of groups on a ticket.

Thus with three groups, 2 to the third power is 8, minus 1 equals 7, the total number of ways.  And with four groups, 2 to the fourth power is 16, minus 1 equals 15, the total number of groups.  Yes, a ticket with 8 groups on it always has 255 ways total!