Years ago, my sister introduced me to a great casual time-wasting game called Farkle!. It’s fairly straightforward, requiring only six dice and some way to keep score. A great way to pass time when you have no power, or you want something casual while waiting for something on vacation.
In a game of Farkle!, players take turns rolling dice. Combinations of dice are worth different numbers of points. As combinations are scored, the dice are set aside and points banked. Choose to roll the remaining dice to keep scoring more points, or stop and keep the points you’ve banked so far. If at any point you roll dice and score no points, you commit the titular Farkle!, giving up all the points made so far that round and passing the dice to the next player. If you score all six dice score (either at once, or over several rolls), you may pick them all up and continue rolling with 6 fresh dice.
First player to 10,000 points wins.
For the uninitiated, here are the scorable [dice] combinations:
- [1] are worth 100
- [5] are worth 50
- Triples (“trips”) are worth 100x dice value (e.g. three [5]s are worth 500)
- X-of-a-kind above triples are worth an additional 100x dice value (e.g. four [5]s are worth 1000, five [5]s are worth 1500, etc.)
Additionally, there are two combinations only possible when rolling 6 dice:
- Three pairs are worth 750
- A six die straight is worth 1250
Exploring Strategy
Like all dice games, Farkle! is mostly about luck. Player strategy comes in when considering how to manage risk. When is a good time to stop and keep what you’ve banked? Does that point depend on how many dice are left? Certainly, fewer dice must mean lower potential returns, and higher likelihood to score nothing and Farkle!, right?
Another avenue for skill expression: players do not have to score every dice possible on each roll. Individual [5]s are quite low value, and may not be worth taking.
This six dice board is worth a maximum of 200 using four dice. But are all four dice worth scoring? Leaving more dice available for your next roll probably results in better odds of scoring something, so maybe it is better to leave more dice on the board. But if true, how many dice to you leave? Do you score only the [1]s and not the [5]s? Do you “stall” as much as possible – only scoring one [1], leaving yourself the best roll odds on your next throw?
Alternatively, does it make sense to score as many dice as possible to try to get back to a fresh set of six dice faster (and access to powerful six-die scoring combinations)?
On that topic, let’s talk trips. They are a great way to score a lot of points. It must be true that rolling trips is easier with more dice available to throw. If you take all four dice above, it becomes impossible to roll trips on your next throw, since only two dice remain. Does that impact your decision?
How much better or worse of a position do these strategies leave you in, specifically? This is the question I want to figure out.
You’ve got two coconuts and you’re banging them together!
A dice game is just a math problem. In particular, this is a combinatorics problem, and answers to these questions, or at least “which decision has the highest expected score?” would undoubtedly be best represented by a series of equations. That sounds too straightforward and not very interesting, so instead I created a digital version of the game Farkle! in Python and simulated millions of games using 378 different strategies. The results of these simulations are explored below.
A starting place: simple distributions of N dice
Before we get into more complex strategy definitions, let’s just look at some simple charts. If you are throwing a certain number of dice, what are you likely to be able to score off of that roll?
Farkle! probability by N dice
One simple – maybe naïve – way to play would be to focus on avoiding Farkle!ing. It stands to reason that the longer you can keep rolling dice, the better off you’ll end up, right? For that strategy, a natural question is “How risky is it to roll given I have N dice?”
| # Dice | Farkle! % |
| 6 | 2.35% |
| 5 | 7.71% |
| 4 | 15.83% |
| 3 | 28.06% |
| 2 | 44.89% |
| 1 | 66.67% |
At 6 dice, you’re pretty unlikely to Farkle!. The sample space, the landscape of possible dice outcomes, is pretty large (720 combinations) and almost all of those outcomes involve scoring at least some points. Scoring nothing here requires a pretty specific result, namely: among the values [2], [3], [4], [6], you must roll exactly two pair of any two of them, and singletons of the other two values.
As we remove dice, we reduce the way of making trips (or more), and reduce the chances of seeing 1s or 5s. This table validates the intuitive feeling that chance to lose goes up with fewer dice. However, it’s hard to contextualize these numbers and use them to form an intelligent strategy. What % chance of loss is too high to be comfortable with? 1 in 5? 1 in 3? Does that depend on how many points you have so far? This information is interesting, but it doesn’t concretely answer our real question.
Defining Player Strategies
Which player strategies are worth exploring? I thought about my own play and the kind of decisions I battled with. I came up with 7 possible axes of strategy that would be interesting to explore:
- Low Value Trips: Does the player take trip/quad [2]s or trip [3]s? These are fairly low-value for the quantity of dice they consume. Maybe its more worth it to avoid low value trips and hang onto more dice, since more dice mean better future rolls.
- Solitary [1]s: How do you treat [1]s? Do you treat them as a consistent value and take any you find, or do you only take them when you have to (like above, to leave more dice available for trips)? In our example below, would you take both [1]s, or only one [1]?
I classify players as either “Take” (always take [1]s), or “Avoid” (only take one [1])
- Solitary [5]s: Similar to above, except with [5]s. [5]s are worth less, so I expect the “Take” position to be much weaker overall.
- Greed One Dice: When left with one dice left to roll, do you go for it or not? Going for it has a 1/3 chance of success (roll a [1] or [5]), and puts you in a powerful position to roll a fresh set of 6 dice.
- Greed Six Dice: If you make it back to six fresh dice, do you go for it? In the section above, we learned that rolling six dice has a pretty low chance of Farkle!ing (2.35%), and the highest odds for large point returns.
- Bank Limit: When do you feel you’ve banked enough points that you should stop rolling? Is 400 enough? 500?
- “Giving up on trips” Transition Point: There comes a time when a “stall” player recognizes they are unlikely to roll trips, should stop “stalling”, and just start taking whatever they are offered. At how many dice left do you start taking all the [1]s and [5]s you see? Clearly when you roll 3 dice you should transition, since after that point you’ll be rolling fewer than 3 dice and will be unable to make trips at all. But maybe its worth transitioning at 4 dice, or even 5. Essentially, this axis is the number of dice at which an “Avoid” player becomes a “Take” player.
Using axes 1-5 I created 9 different players who follow different combinations of these strategies. I branded each player a color for easier identification.
Axes 6 (bank limit) and 7 (transition point) allow for multiple values. You could have many different personal bank limits, and transition on 4, 5, or 6 dice. Rather than make new colors for these axes, I simulated the above 9 colors across many different bank limits and transition points. I chose the following values:
- Bank Limits Tested: [50, 100, 200, 300, 3501My sister and BIL play with a house rule that players cannot bank on 350 points, feeling that this number comes up quite often, and forcing players to go up to at least 400 makes for a better game. I want to test this strategy., 400, 500, 600, 700, 800, 900, 1000, 1500, 2000]
- Transition Points: [3,4,52A transition on 6 player is equivalent to someone with “Take” in both 100s and 50s columns. I didn’t see the need to separate these, since Red and Green players would become identical to a Blue player of the same bank limit, and Blue, Black and yellow players wouldn’t change at all.]
In my data a player can be identified as:
So, Green4_400 is the green player who transitions to “take everything” when down to 4 or fewer dice, and stops once they’ve banked 400 points.
The universal strategies
Some strategies seem obvious. My program assumes every player will always do the following:
- Avoid losing intentionally if scoring a nonzero number points is possible
- Take and score the 6-dice combinations (run-of-six, three-pair)
- Take and score trips+ of [4]s, [5]s, or [6]s, or quads+ of [3]s
Simulating the Players
Each of the 9 players plays 100,000 rounds3“Rounds” here meaning play until the player has decided to bank or has Farkled! of Farkle! using each of the 14 Bank Limits, and each of the 3 transition points, resulting in a total of 37.8MM rounds played. Viewers will note not all of those are really useful. For example the Blue, Black, and Yellow strategies which always take 1s and 5s will behave the same regardless of transition point. However, my CPU worked hard to generate that useless data and deserves to have the full number of rounds reported.
Fun and Messy charts
So, how did our players perform? There is a lot of data here, so let’s look at some graphs. I’ve chosen to focus on the average score of players (# points actually banked). Since players take turns, the player with the highest average # points banked each time it is their turn should reach 10,000 points the fastest and therefore win most often.
How often did they Farkle!, and how many points did they survive to bank?
The non-linear pattern here is interesting. Strategies with low bank limits (small circle size) tend to Farkle! the least often, but because they give up so early they don’t score large numbers of points without lucky six-dice combinations. As bank limits increase, average score increases alongside it up to around 4-500 points. However, as bank limits continue to grow, average score begins to plummet as Farkle! rate increases. The apex of the curve can be seen as a flex point. Strategies to the left of that point are too conservative, while those to the right are too greedy.
Observe some color separation here too. Light Red strategies tend to perform above the curve, with some smaller separation in Red and Light Green as well.
We can see some color and transition point separation here, as well as the same gradual decline in performance as Bank Limit gets too high. Also notice that color strategy performance appears to invert at very small bank limits. Colors that performed the best like reds and greens now perform among the worst. These bank limits are so conservative, they don’t pair well with color strategies that want to stall since they don’t play long enough, and mesh better with strategies that take everything they can get up front (blacks, blues, yellows).
Pink’s dominance is quite clear here, with Light Green and Red pretty closely tied for 2nd place. This view also highlights that the “light” version of each color seems to outperform its counterpart color. The “dark” version tests the strength of Greeding One Dice left (something I liked to do and wanted to validate). Rolling on 1 dice seems to be a poor idea on average.
This passes the sanity test to me. Rolling a fresh set is an opportunity to roughly double your score, but comes with a 66% chance to lose everything. This is a negative expected return. Bad odds.
The Mega Chart
Hideous and beautiful, here it is in all its glory.
Quick Facts and Takeaways
Here is a cleaner tabular view of the color rankings, Bank Limits, and transition points.
Here are some large patterns seen in these tables:
- It is better not to roll on one die left (light strategies outperform).
- Always taking [5]s tends to be bad.
- Transitioning later (i.e. stalling longer) results in more dice rolled, more points banked, and fewer Farkle!s.
- The magic point for bank limit is 300/350. Performance steadily declines after this point, as chance to Farkle! goes up.
So, what is the best strategy?
The best color strategy is Light Red, the strategy that stalls, always rolls 6 fresh dice, and never rolls on 1 die left. It outperforms all other color strategies at every transition point, and at almost every bank limit (exceptions4I leave understanding why as an exercise for the reader. Mail your answers along with a $20 bill to 1600 Pennsylvania Ave, Washington D.C. are 50 and 100).
The best transition point for stalling strategies is 3. In other words: stall for trips as long as possible.
The most consistent bank limit is 300 or 350, with one notable exception: Light Red. The best color strategy actually performs noticeably better between 350-600 limits. My guess is this difference relates to Light Red being a stall for trips strategy which performs a bit better if it is allowed to get that last roll off at 3 dice left. A bank limit of 300 would cause the strategy to stop even with 3 dice left if it took three [1]s previously.
Not tested
Notes on some things I didn’t test that are worth noting.
Differentiating [5]s and [1]s past transition point: Once transitioned from “stall”, my players treated [1]s and [5]s as equally valuable to take. However, its possible that a soft stall strategy, where players continued to stall for [1]s and avoid taking the half-value [5]s unless forced to could continue to provide value between 1-3 dice throws.
Changing strategy based on score of other players: A real player should be willing to change to riskier strategies in the face of imminent defeat. For example, rolling on 1 die left is on average bad, but if your opponent is at 9800 points you should pretty clearly continue to go for it and risk it rather than hand over the dice. My robotic players are stoic and unflinching. They never change strategies, but it would be interesting (and more work for me) to adjust their programming to compensate.
On the other hand, I’ve watched enough football where teams decide to punt in the 4th with 3 minutes left while down 14+, so what do I know. You can only accuse my robots of being as dumb as NFL coaches.
Notes
- 1My sister and BIL play with a house rule that players cannot bank on 350 points, feeling that this number comes up quite often, and forcing players to go up to at least 400 makes for a better game. I want to test this strategy.
- 2A transition on 6 player is equivalent to someone with “Take” in both 100s and 50s columns. I didn’t see the need to separate these, since Red and Green players would become identical to a Blue player of the same bank limit, and Blue, Black and yellow players wouldn’t change at all.
- 3“Rounds” here meaning play until the player has decided to bank or has Farkled!
- 4I leave understanding why as an exercise for the reader. Mail your answers along with a $20 bill to 1600 Pennsylvania Ave, Washington D.C.