Farkle

Farkle (1996) is a dice game, with the goal of scoring the most points by rolling favorable combinations of dice. Players roll 6 dice, choosing to score the dice they rolled, or to continue rolling.

Theory

Expected Value

The key to optimal Farkle strategy lies in the expected value (EV), which is the average outcome you'd expect from a decision if you made it many times. For a random variable $X$ with possible outcomes $x_i$ and probabilities $p_i$ :

\mathbb{E}[X] = \sum_i p_i \cdot x_i

In Farkle, we use expected value to decide whether to bank (keeping your points) or roll (risk points for more).

Farkle Probability

A "Farkle" occurs when you roll no scoring dice. The probability of Farkle depends on how many dice you roll:

Dice	Farkle Probability
1	66.7%
2	44.4%
3	27.8%
4	15.7%
5	7.7%
6	2.3%

With 6 dice, you almost always get something. With 1 die, you're more likely to Farkle than not (only a 1 or 5 scores).

The Roll vs Bank Decision

When you have accumulated $S$ points this turn and scored $s$ points from your current roll (leaving $n$ dice to roll), you face a choice:

Bank: Guarantee $S + s$ points
Roll: Risk losing everything for the chance at more

The expected value of rolling is:

\mathbb{E}[\text{Roll}] = P(\text{score}) \cdot (S + s + \mathbb{E}[n]) + P(\text{farkle}) \cdot 0

where $\mathbb{E}[n]$ is the expected additional points from rolling $n$ dice. You should roll when $\mathbb{E}[\text{Roll}] > S + s$ , which simplifies to:

P(\text{score}) \cdot \mathbb{E}[n] > P(\text{farkle}) \cdot (S + s)

Computing Expected Values

The expected values $\mathbb{E}[n]$ for each number of dice form a system of equations that can be solved iteratively. The solver below precomputes these values using dynamic programming, evaluating all possible dice combinations and their scoring subsets to find the optimal expected value for each game state.

The "Hot Dice" rule (rolling all 6 dice again when all dice score) is especially favorable since there is only a 2.3% chance to Farkle, and continuing is almost always correct.

Copy from BuddyBoardGames

Paste in the browser console on buddyboardgames.com/farkle to add a copy button:

(() => {
    const btn = document.createElement('button');
    btn.textContent = '📋 Copy Dice';
    btn.style.cssText = 'position:sticky;bottom:10px;left:10px;z-index:9999;padding:8px 12px;border-radius:4px;cursor:pointer';
    btn.onclick = () => {
        const r = [...document.querySelectorAll('.die-list')].map(d => d.dataset.roll).join(',');
        navigator.clipboard.writeText(r).then(() => btn.textContent = '✓ Copied!');
        setTimeout(() => btn.textContent = '📋 Copy Dice', 1500);
    };
    document.body.appendChild(btn);
})();

you can then paste (Ctrl+V) the copied dice rolls here.