Complex hearts

Complex Hearts is a variant of Hearts reported to be invented by Richard Garfield. It uses the complex number system for scoring. The rules of play are similar to those of conventional Hearts.

Scoring

Hearts are worth $1+0i$ points apiece.
J♦ is worth $-10+0i$ points.
Q♠ is worth $0+13i$ points.
10♣ multiplies the owner's score for that hand by $0+2i$ .

The loser of a game is the first player whose score, in absolute value, exceeds 100. The winner is the player whose absolute value is smallest. (The absolute value of a complex number $a+bi$ is ${\sqrt {a^{2}+b^{2}}}$ .)

Rules

Rules are similar to those of conventional Hearts, but there are differences.

"Breaking hearts" is done by Q♠, or any heart. J♦ and 10♣ do not break hearts.

Shooting the moon requires all hearts and the Q♠, but the 10♣ and J♦ are optional. It allows the shooter to award $\pm 13\pm 13i$ points to himself or every other player, the sign chosen per player at the shooter's discretion.

The 10♣ may not be played on the first trick, unless the player holding it has no other clubs.

Strategy

In conventional hearts, every point card is undesirable unless the player is attempting to shoot (or prevent shooting of) the moon. Complex hearts adds a new dimension to this. For instance, depending on a player's score, it might be valuable for them to acquire the Q♠. It can be stimulating and fun to try to predict opponents' strategies and thwart them. A player might ask himself: "Hmm. The Q♠ is good for me, because it decreases my negative imaginary score, but it's very bad for Alfred since he took the 10♣. Should I take the Q♠ myself, or drop it on him?"

Typically, trick taking goals break down like this:

If a player has a positive real score, they desire either J♦ without 10♣, or Q♠ with 10♣.
If a player has a negative real score, they desire hearts without 10♣.
If a player has a positive imaginary score, they desire J♦ with 10♣.
If a player has a negative imaginary score, they desire Q♠ without 10♣.

It is worth noting that Q♠ with 10♣ can be a particularly malevolent combination to take, as it yields $-26+0i$ points. The only way to rid one's self of that high magnitude negative real score is to take many hearts. However, it is difficult to take many hearts without also acquiring the J♦, 10♣, or Q♠, all of which can serve to increase the user's score in absolute terms.

Logistics

Because of the relative difficulty of performing complex arithmetic mentally, scoring is best done with a spreadsheet.

References

University of Nebraska, Lincoln