Complex Hearts

Full rules

Three kinds of rule are on this page, each tagged. From the reference marks the variant's rules as reported on the reference page, which attributes the game to Richard Garfield. Our reading marks how we settled something the reference leaves unclear. House rule marks conventions we chose ourselves for the parts the variant does not cover.

Open questions, and our answers

• In what order do the moon, the jack and the ten apply? Card points (with the moon), then the ten, then any moon received.
• Does the ten multiply the jack? Yes: the jack is part of the hand's score.
• Does a shooter's ten multiply the moon? Yes, in both choices; the reference's own example shows it.
• Does a receiver's ten multiply a moon given to them? No: it arrives after their own ten.
• The reference's moon-and-ten figure has the opposite sign to (13 + 13i) × 2i. We follow the multiplication.
• Does reaching magnitude 100 lose, or only passing it? Reaching it.
• What if several players reach 100 on the same hand? They share the loss.
• Who wins? House rule: the non-loser closest to zero.
• Passing, first lead, breaking hearts, the first trick? House rules, listed below.

Scoring a hand

From the reference These values are the variant's, as reported on the reference page:

• Each heart: 1.
• The queen of spades: 13i.
• The jack of diamonds: −10.
• The ten of clubs: whoever takes it has their score for that hand multiplied by 2i.
• Every other card: nothing.

The reference works one example: five hearts, the queen and the ten give (5 + 13i) × 2i = −26 + 10i. Our tests check that this hand scores exactly that.

Shooting the moon. From the reference A player who takes all 13 hearts and the queen of spades chooses one of two things: subtract 13 + 13i from their own score, or add 13 + 13i to every other player's. The reference says the jack and the ten are independent of the moon, so they still count as usual, whoever took them. The jack and ten are not needed to shoot the moon.

Why the ten turns your hand

Multiplying by i turns a point a quarter-turn anticlockwise around zero: 1 becomes i, i becomes −1, −1 becomes −i, and −i becomes 1 again, because i × i = −1. Multiplying by 2i makes the same turn and doubles the distance from zero. In the example, 5 + 13i (magnitude 13.9) turns into −26 + 10i (magnitude 27.9).

The ten turns the hand's score, never your running total. The turned hand score is then added to your total like any other. That is why the Play page shows each hand in two parts: the hand's own working, then the addition.

The order of scoring

Our reading The reference does not list an order. We score each player like this:

1. Add up their card points: 1 per heart, 13i for the queen, −10 for the jack. A moon shooter's hearts and queen are replaced by the moon: −13 − 13i if they subtract, or nothing if they give it to the others.
2. If they took the ten of clubs, multiply the result by 2i.
3. If someone else shot the moon and gave it away, add 13 + 13i last, or (13 + 13i) × 2i = −26 + 26i if the shooter took the ten.
4. Add the hand score from steps 1 to 3 to their running total.

Why this order:

• The jack counts before the ten. The reference says the ten multiplies your score for the hand, and the jack is part of that score. So the jack and the ten together give −10 × 2i = −20i. The reference has no example with both, so this is our reading of its wording.
• The moon is inside the ten. The reference's own moon-and-ten example has the ten acting on the 13 + 13i, in both choices. Multiplying distributes over adding, so it makes no difference whether the moon is subtracted before or after turning: −(13 + 13i) × 2i is the same as −((13 + 13i) × 2i).
• A moon given away arrives last. The reference does not say what happens when one player gives the moon away and another took the ten. We treat the ten as multiplying only what its own taker won, so the moon arrives already multiplied by the shooter's ten or not at all, and no amount is ever multiplied twice. There is only one ten, so this keeps every hand to at most one turn.
• Hearts, queen and jack are simply added, so their order among themselves cannot matter.

A sign slip in the reference's moon example

Our reading For a moon taken together with the ten, the reference page gives 26 − 26i as the amount to subtract from your own score, or to add to everyone else's. But the reference also defines the ten as multiplying by 2i, and (13 + 13i) × 2i = −26 + 26i, not 26 − 26i. We follow the multiplication. Subtracting −26 + 26i moves the shooter by 26 − 26i, the figure the reference prints, so that choice agrees with it as the change in the shooter's score. Giving it away moves each other player by −26 + 26i. Either way the moon is turned and doubled to magnitude 36.8 instead of 18.4, which fits the reference's warning that taking the ten with the moon is usually bad for you.

How the game ends

From the reference According to the reference page, the first player whose score has magnitude 100 loses, where the magnitude of a + bi is √(a² + b²).

• Our reading Reaching counts. A score of magnitude exactly 100 loses, as does anything beyond it. Exactly 100 is possible with whole numbers: 100 + 0i, 60 + 80i, −28 + 96i and others. We check a² + b² ≥ 10,000 using whole numbers, so rounding can never decide a game. A score of 99 + 14i, magnitude 99.98, does not lose.
• Our reading Checked after each hand. Scores change only when a hand is scored, so that is when we check.
• Our reading Reaching it together. If two or more players reach magnitude 100 on the same hand, none of them was first alone, so they share the loss.
• House rule A winner. The reference names only a loser. When the game ends we also name a winner: the player who did not lose and is closest to zero (smallest magnitude). Players at exactly the same magnitude share the win. If every player reaches 100 on the same hand, nobody wins.
• From the reference Who chooses the moon. The shooter; the other players have no say.

Peace Cat Games house rules for play

House rule The variant as reported changes only the scoring and says nothing about how the cards are dealt, passed or played. For those we use common Hearts conventions, chosen by us. They are not Richard Garfield's rules, and other tables may play them differently.

1. Four players, a standard 52-card deck, 13 cards each. Aces are high; there are no trumps.
2. Passing. Before each hand everyone passes three cards: to the left on the first hand, to the right on the second, across on the third, and not at all on the fourth; then the cycle repeats. Everyone chooses before seeing what they will receive.
3. First lead. Whoever holds the 2 of clubs after passing leads it to the first trick.
4. Following suit. You must play a card of the suit led if you have one; otherwise you may play any card, subject to rule 5. The highest card of the suit led takes the trick, and its taker leads the next one.
5. First trick. On the first trick you may not throw a heart or the queen of spades while you hold any other card you could play. The jack of diamonds may be thrown. The ten of clubs is a club, so it can simply be played to follow suit.
6. Breaking hearts. You may not lead a heart until a heart has been played to an earlier trick of the hand, unless you hold nothing but hearts. The queen of spades may be led at any time and does not break hearts.
7. A hand is 13 tricks. It is then scored, and the next hand is dealt, until the game ends.

The computer players

Miso, Pepper and Juniper are short programs that run in your browser. There is no AI service behind them and nothing is sent anywhere.

What they can see. Each is handed only its own hand, the three cards it passed and the three it received, the cards played to each trick and who took them, whether hearts are broken, the scores, and how many cards each player holds. The other hands are not in what they are given, so they cannot peek. The same rule keeps hints and previews on the Play page from showing any card you could not see at a real table.

How they decide. For each trick a computer player asks one question: if I end up taking this trick, how much further from zero does my score go? That accounts for the direction of its own total, so the jack's −10 is welcome only to a player it pulls back towards zero, and the ten counts as turning and doubling whatever it has already won. With that answer it plays ordinary Hearts technique: duck under bad tricks with its highest safe card, win good ones, throw the queen of spades and high hearts when it can't follow suit, lead low, and lead low spades to flush out the queen. It passes the cards most dangerous to hold. If it shoots the moon it works out both choices exactly and picks the one that leaves it best placed.

What they don't do. They look no further ahead than the current trick, don't try to shoot the moon on purpose, don't guess where unseen cards are beyond simple averages, and don't learn from you.

How strong that is. We measured them against two simple strategies, over 400 games each, with one seat played by the strategy and three by the computer, so a fair share of losses would be 25%. A seat playing random legal cards lost 98.3% of games. A seat playing plain Hearts habits that ignore the complex plane (duck when possible, throw the queen and high hearts, lead low) lost 58.5% and won 7.0%. That is all we can say: we have not measured them against people.

Credits and licensing

• Game design attributed to Richard Garfield; the rules first appeared in Games magazine, as reported by Larry Charbonneau's Complex Hearts page, our rules reference.
• Independent implementation (code, card drawings, diagrams and text) by Peace Cat Games; not affiliated with or endorsed by the designer, Games magazine or the reference page's author.
• Rules for dealing, passing and play, and the choice of a winner, are Peace Cat Games house rules, marked as such in the rules.
• License: Implementation © Peace Cat Games, all rights reserved. No claim is made to the game's design or mechanics.

The game. Complex Hearts is a scoring variant of Hearts. Larry Charbonneau's Complex Hearts page, our rules reference, attributes its origin to Richard Garfield and says the rules first appeared in an issue of Games magazine; it does not give the issue, and we have not seen the original article. The design is Richard Garfield's. We claim no ownership of the game or its mechanics.

This implementation. The code, the card drawings, the diagrams and all of the text on these pages were written new by Peace Cat Games. Nothing was copied from the reference page; it was used only as a source for the rules. This is an independent implementation. It is not affiliated with, authorised by or endorsed by Richard Garfield, Games magazine or the reference page's author.

House rules. The rules for dealing, passing and play, the choice of a winner, and our readings of the unclear points are ours, and are marked as such in the full rules above. The original article may settle some of them differently.