# Take another trip over the ‘bridge’

Mar 7, 2005 11:25 PM

The abbreviated bridge system was a powerful tool for Keno checkers, at least those who were "in the know." A great illustration of the tool is the king ticket. Consider an eight spot ticket, grouped all as one spots, eight kings. We know from Pascal’s Triangle that this ticket will have 255 total ways, and of course because all the groups are the same size, we can read the breakout of the ticket right off the Triangle. But let’s suppose that we don’t know about Pascal’s Triangle but we do know the abbreviated bridge system.

We recall from past weeks that we can split a ticket into arbitrary pieces, so a good starting point will be to divide the ticket into two groups of four kings each, 1-1-1-1|1-1-1-1. We know from experience that each one of these halves would break out to a 1 way 4, a 4 way 3, a 6 way 2, and a 4 way 1. This gives us our abbreviated bridge system chart. Remember, we first put our break out halves along the top row and the first column of the chart, and then we multiply the numerators and add the denominators:

 Â 1/4 4/3 6/2 4/1 1/4 1/8 4/7 6/6 4/5 4/3 4/7 16/6 24/5 16/4 6/2 6/6 24/5 36/4 24/3 4/1 4/5 16/4 24/3 16/2

When we add the resultant ways on our chart we find that there are indeed 255 ways on the ticket.

We might find it more convenient to split the ticket into 5 kings/3 kings 1-1-1-1-1|1-1-1. In this case, the five king split will give us in the first column a 1 way 5, a 5-way 4, a 10 way 3, a 10-way 2, and a 5-way 1. On the top row, the three kings will break out to a 1 way 3, a 3-way 2 and a 3-way 1. Let’s construct the chart:

 Â 1/3 3/2 3/1 1/5 1/8 3/7 3/6 5/4 5/7 15/6 15/5 10/3 10/6 30/5 30/4 10/2 10/5 30/4 30/3 5/1 5/4 15/3 15/2

Once again we find 255 total ways, although in a totally different break out.

Well, that’s it for now. Good luck! I’ll see you in line!