Find your ‘way’ with a bridge

Apr 17, 2006 3:58 AM

On more complex tickets, the bridge system is a useful technique for calculating ways. Consider the ticket of 16 spots, grouped 3-3-3-3-1-1-1-1. The bridge system starts by dividing this ticket into two parts, the 3-3-3-3 and the 1-1-1-1, which are temporarily considered to be two separate tickets.

Computing the ways on each ticket, we get a 1 way 12, 4 way 9, 6 way 6 and 4 way 3 on the first one, and a 1 way 4, 4 way 3, 6 way 2, and 4 way 1 on the second one. The ways from the first ticket are listed individually in a column on the left hand side of a sheet of scratch paper, while the ways from the second ticket are listed individually across the top of the scratch paper. The scratch paper should look like this at this stage:

 

4 3 3 3 3 2 2 2 2 2 2 1 1 1 1

12

9

9

9

9

6

6

6

6

6

6

3

3

3

3

 

The next step is to simply sum across the rows and down the columns, using the column and row headers.

 

4 3 3 3 3 2 2 2 2 2 2 1 1 1 1

12 16 15 15 15 15 14 14 14 14 14 14 13 13 13 13

9 13 12 12 12 12 11 11 11 11 11 11 10 10 10 10

9 13 12 12 12 12 11 11 11 11 11 11 10 10 10 10

9 13 12 12 12 12 11 11 11 11 11 11 10 10 10 10

9 13 12 12 12 12 11 11 11 11 11 11 10 10 10 10

6 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7

6 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7

6 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7

6 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7

6 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7

6 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7

3 7 6 6 6 6 5 5 5 5 5 5 4 4 4 4

3 7 6 6 6 6 5 5 5 5 5 5 4 4 4 4

3 7 6 6 6 6 5 5 5 5 5 5 4 4 4 4

3 7 6 6 6 6 5 5 5 5 5 5 4 4 4 4

 

We know by the method described above that this should contain 255 total ways (8 groups) and indeed it does, since the grid above is 16x16 = 256 with one cell missing.

The next step is to simply sum the ways using all elements of the grid including the original left column and top row, and we see that this ticket contains 1/16, 4/15, 6/14, 8/13, 17/12, 24/11, 22/10, 28/9, 36/8, 28/7, 22/6, 24/5, 17/4, 8/3, 6/2, and 4/1, for a total of 255 ways.

The grid itself is useful if hand checking a ticket like this, since it gives a visual display of how each way is constructed.

Well that’s it for this week! If you have any questions or good fortune stories, e-mail me at [email protected] Good luck, I’ll see you in the lounge!