Hearts & the Shoot-the-Moon Calculator
Play Hearts against three opponents, plus the measurement: watching for a moon shot stops three in five, and half of what survives was never stoppable.
Pick three cards to pass, then play a card to each trick. Follow the suit led if you can; hearts cannot be led until one has been discarded. Every heart is a point and the queen of spades is thirteen — unless somebody takes all twenty-six. Everything runs in your browser.
The moon is rare because it gets broken, not because it is hard
Four identical players, none of which ever tries to shoot, played 20,000 hands twice. The only difference between the two runs is one rule: when a single opponent has taken every point so far, take a trick off them if a legal card allows it. No card counting, no inference, nothing else.
| Opponents | Moons | Rate |
|---|---|---|
| never look | 724 in 20,000 | 3.62% |
| watch for the moon | 296 in 20,000 | 1.48% |
The crudest imaginable defence removes 59.1% of them — about three moons in five. That is the finding: the moon is not rare because it is hard to assemble, it is rare because it is easy to break. Switch the opponents above to "never look" and the shot becomes a great deal more available.
And of the 296 that still got through, 156 — 52.7% — were hands where no defender ever held a card that could have taken a trick off the shooter. Not played badly: no legal card, at any point. Those deals were lost when they were dealt.
That same figure of 156 appears in both runs, which is the check that it means what it says: in those hands the defence never has a card to play, so it never changes anything, so the two runs play out identically and the same player shoots. Which leaves 568 moons that were genuinely live — and the rule caught 428 of them, 75.4%. Three in five understates the rule, because it averages over hands where nothing could have been done. Against a live shot, one line of defence is worth three in four.
Both rates describe these particular players and would move with better or worse ones. The ratio is the robust part — the same four opponents either side, differing in one rule. A fresh sample of 1,200 unseen deals is re-run at build time as a check: 37 moons without the rule, 14 with it.
Neither extreme holds. The moon is not a theoretical curiosity that never fires, and watching for it is not a defence that always works.
Trying it is more forgiving than it feels
Twenty-six points are dealt every hand, so an average seat takes exactly 6.5 of them — that is 26 divided by 4 rather than a simulation result, and the run reproduces it to four decimals. A successful shot is therefore worth 19.5 against the table, and an attempt that fails leaving you holding F points needs to succeed (F − 6.5) / (F + 13) of the time to break even.
| A failed attempt leaves you | Break-even success rate |
|---|---|
| 8 points | 7.1% |
| 10 points | 15.2% |
| 13 points | 25.0% |
| 16 points | 32.8% |
| 18 points | 37.1% |
| 20 points | 40.9% |
| 22 points | 44.3% |
| 25 points | 48.7% |
Even a catastrophe that hands you 25 of the 26 points only needs the attempt to work 48.7% of the time. A hand where one player took 20 to 25 — a near miss — averaged 22.4 points in the run, which puts the break-even at 45.0%.
And a half is the ceiling, never reached. Put the whole pack in the formula and it gives 19.5 / 39 = exactly one half — but taking all 26 is the moon rather than a failed attempt, so the worst a real failure can leave you is 25 and the threshold is strictly under a half. The intuition that a failed moon shot is ruinous is wrong, and wrong in a bounded way: any attempt you believe will work more often than a coin flip was worth making.
The formula is exact and involves no policy. Reading a near miss as a proxy for a failed attempt is the one modelling step here, and it is a rough one — nobody in the run was trying to shoot, so these are hands that drifted into most of the points rather than reached for them.
What the deal hands you
Exact hypergeometric arithmetic rather than simulation. Three hearts is the commonest holding, and the long heart suits a shot wants are genuinely scarce:
| Hearts dealt | Chance | |
|---|---|---|
| 0 | 1.3% | |
| 1 | 8.0% | |
| 2 | 20.6% | |
| 3 | 28.6% | |
| 4 | 23.9% | |
| 5 | 12.5% | |
| 6 | 4.2% | |
| 7 | 0.9% | |
| 8 | 0.1% |
Exactly one hand in four holds the queen of spades — 13 cards of 52, no arithmetic needed. What makes her frightening is that 41.6% of the time you hold her you have three spades or fewer, so a couple of rounds of spades forces her out of your hand and onto the table.
And she does not arrive alone. Across the run, whoever took the queen finished the hand with 17.1 of the 26 points on average rather than the 13 she is worth — the player who cannot stop the queen usually cannot stop the hearts either.
How to use
- Choose three cards to pass, then press Pass.
- Play a card to each trick — you must follow the suit led if you can.
- Watch the orange marker: it names anyone holding every point so far.
- Take one trick off them and their moon shot is dead.
- Switch the opponents to "never look" to see how much the watching is worth.
Frequently asked questions
What does shooting the moon mean?
Taking all twenty-six points in a hand — every heart and the queen of spades. Instead of scoring twenty-six you score nothing and all three opponents score twenty-six, so the rule inverts the whole game.
How often does shooting the moon actually happen?
It depends almost entirely on whether anybody is watching, which is more interesting than a rate. Four identical players who never try to shoot produced 724 moons in 20,000 hands with no defence, and 296 with one rule added: if a single opponent holds every point so far, take a trick off them. That is about three in five removed by the crudest defence imaginable.
Are those numbers true of real Hearts?
They describe these particular opponents and would move with better or worse ones, so treat the absolute rates loosely. The ratio is the robust part — it is the same four players either side of the comparison, differing in exactly one rule.
Can a moon shot always be stopped?
No, and this is the part that surprised us. Of the 296 that got through the defence, 156 were hands where no defender ever held a card that could have taken a trick off the shooter at a moment when the shooter was winning it. Not played badly — no legal card, at any point in the hand.
Is trying for the moon worth the risk?
More often than it feels. Twenty-six points are dealt every hand so an average seat takes exactly 6.5, which makes a successful shot worth 19.5 against the table. An attempt that fails leaving you with F points breaks even at (F − 6.5) / (F + 13) — so even a disaster that hands you 25 of the 26 only needs to work 48.7 per cent of the time.
Why is the queen of spades so feared?
Exactly one hand in four holds her, and 41.6 per cent of those times you hold three spades or fewer, so a couple of rounds of spades force her out. She also does not arrive alone: across the run, whoever took the queen finished with 17.1 of the 26 points on average rather than the 13 she is worth.
Does this send anything anywhere?
No. Every hand is dealt and played in your browser, and nothing is uploaded.
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