For my upcoming book Math games with bad drawingsI thought about including a classic dice game called “Drop Dead”. It just missed the cut. Strike one: the lousy name. Strike 2: It is a pure game of chance with no discretion. And strike three: Well, that name again.
Even so, I want to share the game here because it teaches a useful lesson on the math of risk.
When it is your turn You roll five dice and get your total, with one major exception: 2’s and 5 series are fatal. You immediately drop dead and will be removed from the game. Not only that, but Whenever 2s or 5s appear, the other dice are worthless. You will not receive any points for the throw. Then, whether you’ve scored points or not, you Roll any remaining dice againand repeat this process until all five dice have dropped dead. Play for a set number of rounds per player (e.g. four). After that, the highest total number of points wins.
Here is a sample round that lasted six throws and scored a total of 15 points.
Note that I didn’t get any points on my first throw. The # 4 (with a two) negated the collective efforts of The # 1, The # 2, The # 3, and The # 5.
This type of failure is common. Only 13% of the time you score a goal on the opening throw. A single 2 or 5 is enough to spoil the party, and with five potential party spoilers, few parties are left untouched. So you get most of your points with only one or two remaining dice, since smaller “groups” are more likely to succeed.
This leads to our larger topic and the lesson that interests me: Starting with additional dice hardly helps. In fact, there is little use after your eighth die and almost none after your twelfth.
It seems weird. An extra die can’t hurt, right? In the best case scenario, it will increase your score and in the worst case it will show a 2 or 5. At that point, discard it and land right where you started.
Sure, it can’t hurt. But at a certain point it doesn’t help much either. Each die has a 1 in 3 risk of falling dead. Put together many times, this becomes a virtual guarantee: someone will spoil the party. With just twenty dice, the chance of completely avoiding 2s and 5s is only 0.03%, which is roughly your lifetime chance of being struck by lightning.
For example, let’s say you start with 5 quadrillion dice, enough to cover the state of West Virginia. Seems like you should get tons of points right? No Litter by litter, about 1/3 of your dice will spoil the party. This is repeated a hundred times in a row, keeping your score at zero, until you eventually start scoring with just a few dice left. (About 17 points on average.)
5,000,000,000,000,000 cubes. 17 points.
The moral: Don’t design systems where everything has to go right. When your machine is doomed to failure due to a defective part; if your party is spoiled by a late guest; if your game plan collapses when a player leaves his position; then you have a problem. Actually a lot of problems: one per component. Crowds are good for some things, but achieving unanimity is not one of them.
By the way, if you want to turn this into a real game, Joe Kisenwether has a great idea: You can start with as many dice as you want, but your turn ends immediately after your 5th roll. So you want to pick enough dice that you won’t run out of dice (1 or 2 is likely too few), but not so many that you waste early rolls getting zero (so 20 is too many).
Puzzle: What is the optimal number of cubes in this version?