Hearts: Answer to question 13

Q: What is the largest proportion of tricks Alexandra can win in total in all hands played (without Shooting the Moon), and still come first in the game?

A: 166 tricks out of total of 195 played = 85.1%

Firstly, Alexandra wants to score as much as possible, without getting even as much as any other player. By question 1, she can get at most 102 in 16 hands. So she wants to average a score of little over 6 per hand.

Any trick won by the other players should have only scoring cards, to allow her to win as many tricks as possible while accumulating as few points as possible. Thus, at least one trick won by another player in each hand will have the Queen of Spades and three hearts. This would leave 10 other hearts to be won by Alexandra in the other 12 tricks. If another trick was won by another player in any hand, there would be 6 hearts left to be won by her in the remaining 11 tricks. If another two tricks were won by other players in any hand, there would be 2 hearts left to be won by her in the remaining 10 tricks.

After 14 games, the scores are as below:

Notation: #hand (tricks, points) [total scores]
Player Alexandra, Katherine, Michell, Zoë
#1  (12, 10) (1, 16) (0, 0) (0,  0) [ 10,  16,   0,   0]
#2  (12, 10) (0,  0) (1,16) (0,  0) [ 20,  16,  16,   0]
#3  (11,  6) (0,  0) (0, 4) (1, 16) [ 26,  16,  20,  16]
#4  (11,  6) (1, 16) (0, 0) (1,  4) [ 32,  32,  20,  20]

#5  (11,  6) (1,  4) (1,16) (0,  0) [ 38,  36,  36,  20]
#6  (11,  6) (0,  0) (1, 4) (1, 16) [ 44,  36,  40,  36]
#7  (11,  6) (1, 16) (0, 0) (1,  4) [ 50,  52,  40,  40]
#8  (11,  6) (1,  4) (1,16) (0,  0) [ 56,  56,  56,  40]

#9  (11,  6) (0,  0) (1, 4) (1, 16) [ 62,  56,  60,  56]
#10 (11,  6) (1, 16) (0, 0) (1,  4) [ 68,  72,  60,  60]
#11 (11,  6) (1,  4) (1,16) (0,  0) [ 74,  76,  76,  60]
#12 (11,  6) (0,  0) (1, 4) (1, 16) [ 80,  76,  80,  76]

#13 (11,  6) (1, 16) (0, 0) (1,  4) [ 86,  92,  80,  80]
#14 (11,  6) (0,  0) (1,16) (1,  4) [ 92,  92,  96,  84]

To that point, we have averaged just over 11 tricks per game. We cannot win while getting more than 10 tricks in either of the last two games, so it is better to bring the average down only once more. So we finish up after only 15 games.

#15 (10,  2) (1,  4) (1, 4) (1, 16) [ 94,  96, 100, 100]

This solution came from suggestions by Neil Smith and Toby Gottfried.