Threes.

A simple gambling dice game, and a lot of fun. Roll five dice, keep the lowest you can, lowest score wins. A three is worth zero. That's it.

The rules — Chicago style

How a turn works

Roll all five dice. You must keep at least one. The kept dice go into your hand — locked in, contributing to your score. Roll the rest. Repeat until all five dice are locked. Your score is the sum of your hand. Lowest score wins.

Threes are wild zeros. Keep every three you see — they cost you nothing and stay nothing no matter how many you've got.

Shoot the moon (Chicago rule): If all five dice show 6, you shoot the moon and collect 36× the bet from every other active player. It's rare, it's dramatic, and the math on when to chase it is not obvious.

🎲

Five dice

Start with five. Must keep at least one per roll. Locked dice don't move.

✕

Threes = zero

A three contributes nothing to your score. Always keep them. There's never a reason not to.

📉

Low wins

Unlike most dice games, you want small numbers. A score of 0 is a perfect turn.

🌕

Shoot the moon

Five sixes wins 36× the bet per opponent. Pursue it only when the EV math says so.

Expected score from N dice

If you roll N dice and play every subsequent decision perfectly, here's the score you'll add on average. This is the number the tool is constantly comparing against — whenever you consider keeping a die, the question is whether locking it in beats what N more dice would give you.

Dice	Expected score	The key implication
1 die	3.0	You must keep it. No decision — just note whether you got lucky.
2 dice	4.4	Worth keeping any single die valued at 4 or less rather than rolling both fresh.
3 dice	5.2	Worth keeping any subset that totals less than 5.2 — the tool figures out which subset that is.
4 dice	5.8	A very good roll can beat this, but most of the time you let four dice ride.
5 dice	6.3	A full optimal turn averages around 6. Anything under 4 is a good game.

How to use this tool

Analysis mode — set up any position: click a die slot to assign its face, then mark each die as locked (already in hand) or fresh (this roll). The advisor shows what to keep, with exact EV and win probability if you've set an opponent target. Good for reviewing decisions or settling arguments.
Play mode — step through a real turn. Hit roll, see your dice, tap the ones you want to keep. The advisor shows its recommendation alongside yours and tells you how many expected points you're leaving on the table. Turn on "Explain moves" to see the reasoning behind each suggestion.
Opponent target — toggle it on to switch from EV-minimizing to win-maximizing. The optimal keep sometimes shifts when you have a specific score to beat. Sometimes it doesn't — the advisor will tell you.
Moon rules — set the number of other players and the bet size. The advisor calculates moon EV in dollars and tells you exactly when chasing five sixes is mathematically worth it over playing for low score.

Strategy computed via exact dynamic programming — every keep decision is evaluated against the full probability distribution of future rolls. No Monte Carlo, no approximation. The moon probabilities are calculated analytically by chaining the conditional probability of rolling additional sixes through each possible roll sequence.

Position analysis

Click any die to set its face. Toggle between Locked (hand) and Roll (fresh) to describe the position. The advisor updates instantly.

Context

Beat score

Moon rules

Set up a position above — click a die slot to choose its face.

Play mode

Step through a real turn. Tap roll dice to toggle what you keep. The advisor shows its pick alongside yours.