Exploring the Math in 9 Lives

Cat and Calculator - Top View

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.

Comments

Post a Comment

Popular posts from this blog

5 Graphic Design and Typography Tips for your Card Game

Image
Chris Farrell of Illuminating Games just wrote a thorough critique of card games setting their body text to be so small that they cannot be read at arm's length, let alone across the table. Here's an excerpt, but the whole thing is worth a read for any would-be card game designers out there. (Myself included.) Break out your copy of the base set of Dominion, and look at the Chapel. This has a text box roughly 3.5cm by 4.5cm. It's got a single line of text. That line of text is 1 (one) millimeter high. 1mm! For me, it's only clearly readable at half an arm's length even in the bright light of day. (...) Compare this to a more sane game like Glory to Rome, where the font size is 2-3 times as large (text is 2mm high, keywords 3mm and usually highlighted). I can generally read Glory to Rome cards across the table, and can certainly see the important keywords. To heap insult upon injury, not only does Glory to Rome have far more legible text than Dominion it also has
Read more

Low-High Dice Game

Image
A dice game inspired by the stock market (and playing lots of Martian Dice). Roll a bunch of dice and choose which sets to keep. Choose your strategy wisely! Stuff You Need Two-to-Six PLAYERS A PENCIL and PAPER to keep score Thirteen standard six-sided DICE How to Play The shortest player takes the first turn. On your turn, first roll all thirteen dice. Several dice will have matching results. These are called SETS. (A single die result is a set, too.) You must choose a set to keep. For example, your first roll results are 111124445555 . The sets are four 1 s, one 2 , three 4 s, and four 5 s. After keeping a set of dice, lock them up in a row. This is called the GOOD TRACK. For example,  you could keep the set of four 1 s, one 2 , three 4 s, or four 5 s . You decide to keep the four 1 s and line them up in your good track. After keeping a set, you may end your turn or re-roll the remaining unlocked dice. When you re-roll, immediately set aside any results that ar
Read more

The Thousand-Year Game Design Challenge

Image
UPDATE: The submission period has closed and here are the entries. If you're an entrant, continue sending pics or vids of people playing your game. The first step to a game lasting a thousand years is for people to play it today. UPDATE 2: And here's the winner! To support games designed for longevity – that can be learned, played and shared for hundreds of years – we offer this challenge to any game designers, artists and imaginative people who also share this desire. Challenge Create a game. The game can be of any theme or genre you desire, but there is one restriction: You're creating a "new classic," like Chess, Tag or card games. So, create a game to be enjoyed by generations of players for a thousand years. Prize $1,000 to the winning entrant, to be announced and awarded January 1, 2012. Entry Deadline Entries must be submitted before midnight July August 31st, 2011. ( Update ) Entry Guidelines Enter by posting a comment below with a link
Read more