### Exploring the Math in 9 Lives

Last week I posted a simple casino-style game called 9 Lives. I noticed some peculiar math behind the mechanics. I bet this is one of those things mathfolk already know intuitively, but coming at it in long-hand on my notebook over a cup of coffee is still worthwhile.

To review, the game involves playing pairs of cards, each showing a digit, either 0 through 9. Added together, the highest pair is the winner for that turn. The winner earns points equal to the 'ones' digit of their play. So if you played a 12 and won, you'd win 2 points. (Tied players both score.)

So I wrote out the different pairs of digits that would make each sum, from 0 through 18. I also compared that to the value of each sum. This produced the following chart. (Click to enlarge.)

There is only one way to make 0, 1, 17, or 18. Two ways to make a 2, 3, 15, or 16. Three ways to make a 4, 5, 13, or 14. Four ways to make a 6, 7, 11, or 12. Five ways to make  8, 9, or 10.

This makes scoring strategy very peculiar indeed. From 0-9 it's easier to make a winning pair and you get more points for each pair up to 9.

Then you fall off a cliff drop at 10. Above 10, value and difficulty have an inverse relationship. It gets harder to make pairs 10-18, the points earned start again from 0, only reaching 8 at a maximum. Is that a bug or a feature? All I can say for now is that it's a prisoner's dilemma.

Everyone knows optimal play is 9. It's the easiest pair to make and worth the most points in the game. You could pursue that, even if it means sharing the top spot with another player. Or you could go for a higher pair, even if it means you will score fewer points. At least that way, only you will score points. But why ever play 10? It beats anything from 0-9, but scores the winner nothing. Spite?

It also makes me wonder if I should add some deeper auction element, to offer some long-term set-building. Of course, that's my usual go-to solution, but it's a place to start.

The highest player earns first dibs from amongst all the cards in play this turn. She collects one and adds it to a private tableau. She is followed by the next highest player, and so on, until each player has collected one card. The remainder are shuffled back to the bottom of the deck. At the end of the game, bonus points are scored for variety and for sets-of-a-kind.

So you still have a consistent, evergreen reason to pursue a high pair above 9. Plus, even if you don't get first pick at the auction, you can at least get some points as compensation.