# Another way of counting ways

Dec 25, 2006 2:43 AM

There is another slightly faster method of calculating the ways on way ticket than the one we discussed last week. This method is called the Bridge System. It came into use in the 1960’s. When I say that it is faster, I really mean that it is faster with a pen or pencil; last week’s method is more suited for electronic computers.

If you remember, the ticket we chose to illustrate last week’s method was the 15-spot ticket grouped 4-4-4-1-1-1. We discovered that the ticket had 63 total ways.

We can use the same ticket this week.

The first step in using the bridge system is to split the ticket into two separate partial groupings. Usually the best way to do this is to put roughly half of the groups into each grouping. Here we would use 4-4-4 for one grouping, and 1-1-1 for the second grouping.

The second step is to treat each partial grouping as a separate ticket, and calculate the ways for each grouping. With knowledge and experience we can normally "eyeball" the solution for each partial grouping. Here we can see that the 4-4-4 grouping contains a 1-way-12, a 3-way-8, and a 3-way-4, while the 1-1-1 grouping contains a 1-way-3, a 3-way-2, and a 3-way-1.

The third step in the bridge system is to create a simple chart, using all the ways of the first partial solution vertically, and the results of the 2nd partial solution horizontally:

 3 2 2 2 1 1 1 12 8 8 8 4 4 4

The fourth step is to take the chart above, and simple add together the numbers at the top of each column to the numbers at the beginning of each row:

 3 2 2 2 1 1 1 12 15 14 14 14 13 13 13 8 11 10 10 10 9 9 9 8 11 10 10 10 9 9 9 8 11 10 10 10 9 9 9 4 7 6 6 6 5 5 5 4 7 6 6 6 5 5 5 4 7 6 6 6 5 5 5

We can see by inspection that there are 7 x 7 + 7 + 7 or 63-ways total on the ticket.

The last step in the bridge system is to combine all the like ways together, giving us a 1-way-15, 3-way-14, 3-way-13, 1-way-12, 3-way-11, 9-way-10, 9-way-9, 3-way-8, 3-way-7, 9-way-6, 9-way-5, 3-way-4, 1-way-3, 3-way-2 and a 3-way-1.

Well, that’s it for this week, good luck, I’ll see you in line! email: [email protected]