What a ‘way’ to go!

Mar 29, 2005 7:25 AM

One of my all-time favorite tickets is marked with 17 spots, grouped 3-3-3-2-2-2-2. (You may notice that many of my favorite tickets have groups of threes and twos on them, and of course two and three equals five, my favorite number to play!)

Instead of using the bridge system to determine the ways on this ticket, let’s turn to a less formal, but quicker method called the rule of the twins. You will note that with 17 numbers, 16+1=17, 15+2=17, 14+3=17, 13+4=17, 12+5=17, 11+6=17, 10+7=17, and 9+8=17.

This being true, it is also true that we’ll find the same number of way 12s as way fives on the ticket, or the same number of way 10s as way sevens, etc.

This has to be true because if you take a five spot off of this ticket (there is a 12-way 5) this leaves a 12 spot. Thus there is not only a 12-way 5 but a 12-way 12 spot.

The easiest way to use the rule of the twins is to start a chart with the obvious one-way 17:

1/17

Now, below this and to the right, write down the also obvious four way two:

1/17

 
 

4/2

Since 17=2+15, we can see that there will be the same number of 15s as twos, so below the 17 and to the left of the 4/2, we write 4/15:

1/17

 

4/15

4/2

We’ll continue this process until it’s complete:

1/17

 

4/15

4/2

3/14

3/3

6/13

6/4

12/12

12/5

7/11

7/6

18/10

18/7

13/9

13/8

We know from Pascal’s Triangle that there will be 127 ways on the ticket, and indeed there are. The advantage of using the rule of the twins over the bridge system is that it is faster, especially if you only want to determine the numbers of one or two ways.

One of my favorite ways to play this ticket is to play the 7 way 11.