Hearts: Answer to question 20

Q: Provide a complete history of all scores for all players in each hand, if
(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.
(e) There were 5 single hand scores of 18, not necessarily the same player each time.

A: The following game satisfies all the above criteria.

Notation: #hand (points) [total scores]
Players: Alexandra, Katherine, Michelle, Zoë
#1  ( 2,  4,  7, 13) [ 2,   4,   7,  13]
#2  ( 3,  3,  2, 18) [ 5,   7,   9,  31]
#3  ( 2,  4, 14,  6) [ 7,  11,  23,  37]
#4  ( 2,  6, 14,  4) [ 9,  17,  37,  41]
#5  ( 2,  2,  4, 18) [11,  19,  41,  59]
#6  ( 2,  4, 18,  2) [13,  23,  59,  61]
#7  ( 3, 18,  2,  3) [16,  41,  61,  64]
#8  ( 3,  2,  6, 15) [19,  43,  67,  79]
#9  ( 4, 16,  4,  2) [23,  59,  71,  81]
#10 ( 2, 14,  8,  2) [25,  73,  79,  83]
#11 ( 4,  6,  2, 14) [29,  79,  81,  97]
#12 (18,  2,  2,  4) [47,  81,  83, 101]