Learn to count the way (tickets)

July 01, 2002 11:20 PM
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When way tickets start to get complicated, calculating the possible ways to play on the ticket can start to get complicated too. Even relatively simple tickets like we have dealt with in the last few weeks can be difficult to compute.

Here is one simple way to calculate the ways on any ticket. Suppose we have a ticket marked with 15 spots, grouped 4-4-4-1-1-1. The first step is to take the first group and write it down:

4

The second step is to add the second group to the first, and copy the results:

4+4=8

The third step is to add the third group to the results above, and copy the total results to the column:

4+4=8+4=12

We then add the next group to the results, and proceed as above, adding the group of 1 to all previous results:

4+1=5

4+1=5

8+1=9

4+1=5

8+1=9

8+1=9

12+1=13

Next we follow the same procedure with the 2nd group of 1:

4+1=5

4+1=5

8+1=9

4+1=5

8+1=9

8+1=9

12+1=13

1+1=2

5+1=6

5+1=6

9+1=10

5+1=6

9+1=10

9+1=10

13+1=14

For the last step, we take the final group of 1 and add it to all previous results:

4+1=5

4+1=5

8+1=9

4+1=5

8+1=9

8+1=9

12+1=13

1+1=2

5+1=6

5+1=6

9+1=10

5+1=6

9+1=10

9+1=10

13+1=14

1+1=2

5+1=6

5+1=6

9+1=10

5+1=6

9+1=10

9+1=10

13+1=14

2+1=3

6+1=7

6+1=7

10+1=11

6+1=7

10+1=11

10+1=11

14+1=15

Now, having completed this simple task of repetitive addition, we can see that there are 63 ways on this ticket, to wit: 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 11, a 1-way 3, a 3-way 2 and a 3-way 1.

Simple, eh?

Let’s see if we can make it even simpler.

CROSSING THE BRIDGE

There’s a slightly faster method of calculating the ways on a way ticket. 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.

PARTIAL GROUPINGS

Now that we know how to "break out" the ways, as a keno writer would say, using the bridge system, we’ll examine why it doesn’t matter which partial groupings are used. Any may do.

Let’s take the 12-spot ticket, grouped 3-3-2-2-1-1. If we take half the groups for each partial grouping to initiate the bridge system, we’ll get a 3-3-2 and a 2-1-1. Although this partial grouping is usable, it may not be the easiest one to use, especially on more complex tickets.

So, let’s use the partial groupings 3-3 and 2-2-1-1. It’s much easier to eyeball the breakouts of these partial groupings, to wit: 3—3 = a 1-way 6 and a 2-way 3, while 2-2-1-1 = a 1-way 6, 2-way 5, 3-way 4, 4-way 3, 3-way 2 and a 2-way 1. Charting these ways like last week, we get:

0 6 5 5 4 4 4 3 3 3 3 2 2 2 1 1

6 12 11 11 10 10 10 9 9 9 9 8 8 8 7 7

3 9 8 8 7 7 7 6 6 6 6 5 5 5 4 4

3 9 8 8 7 7 7 6 6 6 6 5 5 5 4 4

And this chart displays all 63 ways on the ticket.

We can alternatively use the partial groupings 3-3-2-2 and 1-1 where 1-1 = 1-way 2 and 2-way 1 and 3-3-2-2 = a 1-way 10, a 2-way 8, a 2-way 7, a 1-way 6, a 4-way 5, a 1-way 4, a 2-way 3 and a 2-way 2. Charting this, we get the alternative breakout:

10 8 8 7 7 6 5 5 5 5 4 3 3 2 2

2 12 10 10 9 9 8 8 8 8 8 6 5 5 4 4

1 11 9 9 8 8 7 6 6 6 6 5 4 4 3 3

1 11 9 9 8 8 7 6 6 6 6 5 4 4 3 3

which contains the same ways as the first breakout in a different configuration.

The moral is that, when using the bridge system to break out a way ticket, you can choose the most convenient format.

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