Farkle Cheat Sheet
Farkle has infinite EV and does not converge. With a practical cutoff at 11000 points, I had codex solve for optimal play including the value of 6 fresh dice, which is the most important heuristic for play. Derived from exact dynamic programming under one ruleset. (Note: our rules)
Assumptions Used
- Turn-level optimization: maximize expected points this turn.
- Hard cap for recursion at 11,000 turn points (20,000-point recursion limit resulted in negligible difference).
- Scoring: single 1 = 100, single 5 = 50.
- 3-of-a-kind: 3x1 = 300, 3x(2..6) = face x 100.
- 4-kind = 1000, 5-kind = 2000, 6-kind = 3000.
- Straight (1-6) = 1500, three pairs = 1500, two triplets = 2500, four-kind+pair = 1500.
Core Results
- Fresh turn EV from
(t=0, n=6): U(0,6) = 542.063240. - Farkle probabilities:
P(Farkle|n=1)=4/6=0.666667,P(Farkle|n=2)=16/36=0.444444,P(Farkle|n=3)=60/216=0.277778.
Stop vs Roll Cutoffs (Dice Remaining)
Stop at or above these totals; otherwise roll.
| Dice Remaining | Stop Threshold |
|---|---|
| 1 | 282+ |
| 2 | 234+ |
| 3 | 393+ |
| 4 | 948+ |
| 5 | 2766+ |
| 6 | No cutoff below 11,000 (roll) |
When to Steal
Decision: steal opponent's banked stack with handed dice (must roll once) vs decline and start fresh (fresh EV = 542.063240).
| Handed Dice | Minimum to Steal |
|---|---|
| 1 | 1130 |
| 2 | 799 |
| 3 | 601 |
| 4 | 466 |
| 5 | 319 |
| 6 | 0 |
Key Example
Takeover state (t=1000, n=2): R(1000,2) = 652.766739.
Simple Definitions
U(t,n): best expected final score from state (t points at risk, n dice left), with stop/roll choices allowed.R(t,n): expected final score from state (t,n) if you must roll right now.C(x)=U(x,6)-x: continuation premium for six dice at total x.
1-Die Threshold Equation
With 1 die left:
R(t,1) = (1/6)U(t+100,6) + (1/6)U(t+50,6)
and
R(t,1) - t = -(2/3)t + 25 + (C(t+50)+C(t+100))/6
At around t = 282, this lands near zero, which is why 282 is the practical 1-die stop point in this model.