There are many interesting questions which could be asked about possible games of the card game Hearts. All versions of Microsoft Windows have this game. Here are some (with links to answers):

1. What is the highest possible game score for one player?
2. Can all players tie at the end of a game? Show the scores if they can, or explain why not.
3. What is the highest first place score, if no first place ties are allowed?
4. What is the highest second place score, if no second place ties are allowed, and no one Shoots the Moon?
5. What is the highest third place score, if no third place ties are allowed, and no one Shoots the Moon?
6. What is the largest winning margin? That is, what is the maximum gap between first and (perhaps equal) second at the end of a game?
7. What is the largest losing margin? That is, what is the maximum gap between last and (perhaps equal) second last at the end of a game?
8. I get the same score in each hand of a game, in which I come last. The other three players have the same score at the end of the game. Given that my score in each hand is as low as possible, what are all the scores?
9. I get the same score in each hand of a game, in which I come first. The other three players have the same score at the end of the game. Given that my score in each hand is as high as possible, what are all the scores?
10. I win the same number of point scoring cards in each hand, and come last in the game. How small could that number of point scoring cards be, if my final score is less than 20 more than the winner's score?
11. What is the maximum number of hands in a game if I win the Queen of Spades in every hand and I win the game?
12. What is the smallest proportion of tricks I can win in total in all hands played, and still come last in the game?
13. What is the largest proportion of tricks I can win in total in all hands played (without Shooting the Moon), and still come first in the game?
14. What is the minimum gap between first and last that is not exceeded, after the second hand, for a whole game?
15. I walked in on a game just after the last hand had been scored. When I saw the scores, which were in arithmetic progression, I knew that someone had Shot the Moon in the last hand - and I knew who it was. What could the winner's score be?
16. In another game in which the scores were in arithmetic progression, only the loser's score was not prime. What was the winner's score?
17. In another game in which the scores were in arithmetic progression, only the winner's score was not prime. What was the winner's score?
18. In another game in which the scores were in arithmetic progression, only one score was not a square. What was that non-square score?
19. In yet another game, the scores were in geometric progression. What was the winner's score?
    
    ...and some much harder questions...
    
    I wanted to find some questions about complete histories of games, solved with the minimum of information. I used a form of brute force to find these. The following are games where, in each case:
    (a) Alexandra was always ahead of (had a lower progressive game score than) Katherine, who in turn was always ahead of Michelle, who in turn was always ahead of Zoë.
    (b) progressive scores for all players throughout the game were either primes or squares
    (c) all players scored more than one in each hand.
    (d) the game lasted as many hands as possible under the above conditions.
    
    In each case, you must provide a complete history of all scores for all players in each hand.
    
    I have no easy solution method for these questions (there are 780 different games, many differing in only two hands). I would be pleased and very impressed if you could come up with a neat method to find any of them.
    
20. In one game, there were 5 single hand scores of 18, not necessarily the same player each time.
21. In another game, there were 3 single hand scores of 18, and 18 single hand scores of 2, not necessarily the same player each time.
22. In another game, there were 5 single hand scores of 14 and just one single hand score of 9, not necessarily the same player each time.

Remember:

• in each hand, the total score for all four players increases by either
  • 26 when more than one player wins point scoring cards, OR
  • 78 = 3 x 26 when one player 'Shoots the Moon' - wins all the points scoring cards.
• the game finishes as soon as one or more players' scores exceed 100.
• for Windows users, full rules may be found in the Hearts help file.

Please send me any comments, corrections, improved solutions (I will give full attribution for any I use here).

*** Thanks to Toby Gottfried and Neil Smith for getting me thinking about questions 15 to 19, and for improvements to a number of answers. ***

All text is copyright © Mark Michell 2002, but may be used freely for non-profit purposes. Write to me about any other uses of these puzzles. It would be nice of you to say where you got this if you do use it elsewhere.

This page last edited on 01 December 2002