Let me count the (keno) ways, part 2

January 23, 2001 10:13 AM
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There’s a slightly faster method of calculating the ways on a way ticket than the one we discussed last week. It’s called the bridge system. It came into use in the 1960s. When I say it’s faster, I really mean it’s faster with a pen or pencil. Last week’s method is more suited for electronic computers.

If you remember, the ticket we chose for last week’s method was the 15-spot ticket, grouped 4-4-4-1-1-1. We discovered it 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 the groups into each grouping. Here we’d 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. The 1-1-1 grouping contains a 1-way 3, a 3-way 2, and a 3-way 1.

The third step is to create a simple chart, using all the ways of the first partial solution vertically, and the results of the second 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 simply 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 is to combine all the like ways together, giving us a 1-way 15, a

3-way 14, a 3-way 13, a 1-way 12, a 3-way 11, a 9-way 10, a 9-way 9, a 3-way 8, a

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

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