Zombie Dice Game & Optimal Stopping
Play the push-your-luck dice game, and see the exactly solved stopping rule — which depends on how many red dice are left, not just on how many brains you hold.
Three regimes, and only one of them is a decision
There is exactly one choice in Zombie Dice — roll again or bank — taken repeatedly. Solving the game exactly over every reachable position gives an answer that splits cleanly by how many shotguns are showing:
| Shotguns showing | What to do |
|---|---|
| 0 | Always roll. The solved threshold is 50 brains, and the game ends at 13 — so it can never be reached. |
| 1 | Stop between 6 and 10 brains, depending on the cup. This is the only real decision in the game. |
| 2 | Bank as soon as you have a single brain, whatever the cup looks like. |
Two of the three regimes are not judgement calls at all. With nothing showing you press on; with two shotguns you take what you have. Everything anyone could argue about lives in the middle row.
A shotgun from a red die is good news
Red dice carry three shotgun faces against green's one, so they are the dangerous ones. With a single shotgun already showing, the exact threshold depends on how many reds are still in the cup to draw:
| Cup remaining | Reds left | Stop at |
|---|---|---|
| 6 green, 4 yellow, 2 red | 2 | 6 brains |
| 5 green, 3 yellow, 1 red | 1 | 8 brains |
| 3 green, 2 yellow, 0 red | 0 | 10 brains |
Clearing the reds out of the cup nearly doubles how far you should push — 6 brains up to 10. That runs against the instinct that a shotgun means be careful. If the shotgun came off a red die, that die has left the cup, and what you draw from next is measurably safer than it was. The right response is to press harder.
The same logic cuts the other way for green. A shotgun from a green die removes a safe die and leaves the cup more dangerous, so the threshold falls to 5 — against 6 for a red. Same number of shotguns, different advice, decided entirely by which colour died.
The dice, and what a turn is worth
| Die | How many | Brains | Footprints | Shotguns |
|---|---|---|---|---|
| green | 6 | 3 | 2 | 1 |
| yellow | 4 | 2 | 2 | 2 |
| red | 3 | 1 | 2 | 3 |
13 dice in all. Played optimally from a fresh cup a turn is worth 2.2591 brains, computed exactly rather than simulated — so thirteen takes about six good turns, and the distance between that average and any particular turn is most of what makes the game fun.
Two things the solver got wrong before it got them right, both worth recording. Rolling three footprints returns you to an identical position — same cup, same hand, same shotguns, same brains — so a naive recursion never terminates. It is handled algebraically instead: if rolling returns to the same state with probability q and everything else contributes E, the position is worth E divided by (1 − q). And the first comparison of "which colour was the shotgun" held the cup fixed across all three colours, which made the rows identical by construction rather than by discovery. Reducing the cup by the die that actually produced the shotgun is what uncovered the effect above.
How to use
- Press Roll to draw three dice from the cup and throw them.
- Brains are banked for the turn, shotguns accumulate, footprints get rerolled.
- Three shotguns ends the turn with nothing at all.
- Bank whenever you like — the page shows what the solved policy would do.
- Reach thirteen brains to win.
Frequently asked questions
When should you stop rolling in Zombie Dice?
It depends entirely on how many shotguns are showing. With two, bank the moment you have a single brain. With none, keep rolling — the solved policy does have a stopping point but it sits at fifty brains, and the game ends at thirteen, so it is unreachable. Everything worth deciding happens at exactly one shotgun.
What is the threshold at one shotgun?
Between five and ten brains, and which end you are at is decided by the cup rather than by you. With two red dice still in the cup the threshold is six; with one red left it rises to eight; with the reds gone entirely it reaches ten. Drawing the reds out nearly doubles how far you should push.
Why does the colour of the shotgun matter?
Because the die that produced it has left the cup. A shotgun from a red die removes the most dangerous die in the game, leaving the cup measurably safer than it was, so the right response is to press harder rather than more carefully. A shotgun from a green die removes a safe one and drops the threshold to five, the lowest anywhere.
How many brains is a turn worth?
About 2.26 played optimally from a fresh cup, computed exactly rather than simulated. Thirteen brains therefore takes roughly six good turns, and the gap between that average and what actually happens on any given turn is most of what makes it a party game.
What are the dice?
Thirteen in all: six green, four yellow and three red. Each has six faces divided between brains, footprints and shotguns — three, two and one for green, an even two apiece for yellow, and one, two and three for red. So a red die is three times as likely to end your turn as a green one.
How was the stopping rule worked out?
By solving the game exactly with a dynamic program over every reachable position: the cup composition, the dice in your hand, the shotguns showing and the brains banked. No simulation is involved in the thresholds, and every figure quoted on this page is re-derived from that solver by the tests rather than stored and trusted.
Is it ever right to keep rolling with two shotguns?
Only with no brains banked, when you have nothing to lose. From one brain onward the answer is to stop, and that holds for every cup composition we checked. One more shotgun takes everything, and three dice give it too many chances.
Does this send anything anywhere?
No. The dice are drawn and rolled entirely in your browser, and nothing is uploaded.
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