How these numbers were worked out
Every figure here comes from counting, not from experience. For the odds, a computer goes through every possible throw of one to six dice (46,656 of them for six dice) and checks which ones score. For the banking table, it works out, for every number of dice and every turn total, the expected points of rolling on, choosing the best dice to keep each time, and compares that with banking now. The scoring is the standard chart on the rules page. The computer player and the Hint button in the game use the same table.
The odds of farkling
| Dice thrown | Chance of a farkle | Exactly | Chance every die scores (hot dice) |
|---|---|---|---|
| 6 | 2.3% | 5/216 | 7.8% |
| 5 | 7.7% | 25/324 | 3.0% |
| 4 | 15.7% | 17/108 | 4.0% |
| 3 | 27.8% | 5/18 | 5.6% |
| 2 | 44.4% | 4/9 | 11.1% |
| 1 | 66.7% | 2/3 | 33.3% |
Six dice almost never farkle, and hot dice are likelier with six dice than with five, four or three, thanks to straights and three pairs. One die is a special case: it farkles two times in three, but the one time in three it scores, it is hot dice and you throw all six again.
When to stop rolling
This table gives the turn total at which banking beats rolling on, for each number of dice you have left to throw. It is the strategy that scores the most points per turn on average, about 600.
| Dice left to throw | Bank with this turn total |
|---|---|
| 6 | never: always roll |
| 5 | 3,100 or more |
| 4 | 1,050 or more |
| 3 | 450 or more |
| 2 | 300 or more |
| 1 | 350 or more |
Two things stand out. With six dice, after hot dice, you should always throw again: the chance of a farkle is too small to matter. And one die is worth throwing for longer than two dice, because the one time in three that it scores gives you six fresh dice.
Which dice to keep
You only have to set aside one scoring die, and keeping fewer dice is often better than keeping every point on the table. Some results from the same calculation, at the start of a turn:
- Throwing 1, 5, 2, 2, 3, 6: keep only the 1 and throw five dice. That is worth about 404 points on average, against 311 for keeping both the 1 and the 5 and throwing four.
- Throwing 5, 5 and four dice that do not score: keep one 5, not both (about 369 against 280).
- Throwing 3, 3, 3, 1, 4, 6: keeping only the 1 is worth about 404, a shade more than the three 3s and the 1 together (400), and far more than the three 3s alone (332).
Late in a turn the answer changes. With 400 points already and four dice showing 1, 5, 2 and 3, keep both the 1 and the 5 and bank 550.
Playing to win, not just to score
Scoring the most points per turn is not quite the same as winning. When you are far behind near the end, you need to take more risk; when you are well ahead, less. In the final round you simply need to beat the leader, whatever the odds. A strategy that maximises the chance of winning, for one set of rules, has been computed (Busche and Neller, Optimal Play of the Farkle Dice Game, 2017), but it depends on both scores and is too large to memorise. The banking table above is the practical version.
Rules of thumb
- With six dice, always throw.
- With five dice, keep throwing unless you already have a very big turn (3,100 or more).
- With four dice, bank from about 1,050; with three, from 450; with two, from 300; with one, from 350.
- Early in a turn, keep a single 1 or a single 5 and throw the rest, rather than every scoring die.
Frequently asked questions
What are the odds of a farkle?
With six dice 2.3%, five 7.7%, four 15.7%, three 27.8%, two 44.4% and one 66.7%.
When should you stop rolling in Farkle?
Bank once your turn reaches about 1,050 with four dice left, 450 with three, 300 with two or 350 with one. With five dice keep throwing unless the turn is very large, and with six dice always throw.
What are the odds of farkling with 2 dice?
44.4%, exactly 4 in 9: of the 36 ways two dice can land, 16 have no 1, no 5 and so nothing that scores.
Should you keep every scoring die in Farkle?
Not early in a turn. Keeping a single 1 and throwing five dice is usually worth more than keeping a 1 and a 5 and throwing four.
What is the best Farkle strategy?
To score the most per turn: follow the banking table on this page and keep few dice early. To win, also take the scores into account: take more risk when behind and in the final round.