Hearts: Answer to question 14

Q: What is the minimum gap between first and last that is not exceeded, after the second hand, for a whole game?

A: 7

For example, progressive game scores may go as follows:

                            Maximum
Hand / Progressive scores   difference
#1       5    5    3   13     10
#2      10   10   16   16      6
#3      23   21   17   17      6
#4      25   25   30   24      6
#5      31   31   31   37      6
#6      44   38   37   37      7
#7      44   44   44   50      6
#8      57   51   50   50      7
#9      57   57   57   63      6
#10     70   64   63   63      7
#11     70   70   70   76      6
#12     83   77   76   76      7
#13     83   83   83   89      6
#14     96   90   89   89      7
#15     96   96   96  102      6

By the end of the 6th hand, a difference of 6 always becomes unsustainable. A maximum gap of 7 is easy to sustain.

(I am interested in any more rigorous explanations of this one.)