A king ticket is a keno way ticket that is comprised wholly of kings (1-spot groups). Although this ticket is notorious for its difficulty in checking among keno writers and checkers, a little bit of knowledge can go a long way in easing this process.
King tickets are a special case of the same group way ticket; unique in the fact that every number works with every other number to create the ways. The keno formula may be applied to king tickets to determine the number of ways, but it is somewhat cumbersome. A quicker approach is Pascal’s Triangle, developed by the mathematician Blaise Pascal several centuries ago in the study of probability theory. Pascal’s Triangle is started by writing the numbers 1 and 1 at the top of a sheet of paper.
Below them, and starting to the left, repeat the 1, and sum diagonally below any numbers above, and continue until a final 1 is written to the right.
This process may be continued indefinitely, and it provides a quick visual representation of all the ways on king tickets. On the second horizontal row, the ticket with two kings is represented, with a 1-way-2 and 2-way-1 represented. (The final 1 is ignored, it represents a 1-way-0.) On the third row, a 1-way-3, a 3-way-2 and a 3-way-1 is represented. On the fourth row, a 1-way-4, 4-way-3, 6-way-2, and 4-way-1 is represented. The fifth and further rows follow the same process with the same results.
It is interesting to note that Pascal’s Triangle may actually be applied to any same group ticket, if groups of different size are substituted for the kings. For instance, if groups of three are used, the fifth row of Pascal’s Triangle would represent 1-way-15, 5-way-12, 10-way-9, 10-way-6, and a 5-way-3.
King tickets pack the maximum number of ways into the minimum number of spots, and this is what causes the difficulty in checking them. But it is also true that although a king ticket may have a multitude of ways, it only has a number of distinct catches that is one greater than the number of kings. For instance, a king ticket with eight kings, though it has 255 total ways on it, only has nine distinct catches on it. Thus, it is only necessary to work out the nine catches on it to totally check the ticket. Techniques exist to make this task easier, which I may cover in future columns.
If you have a keno question that you would like answered, please write to me care of this paper, or contact me on the web via email at [email protected]. Well, that’s it for now. Good luck! I’ll see you in line!