According to the conditions of the puzzle
handshakes were only between the men. Thus, the total number of the
handshakes is defined by the number of men arrived. If there were two
men in total, then only one handshake would be performed. If there
were three men, then three handshakes would be performed between them.
If there were four men, six handshakes would be performed. Thus, there
were four men in total - one host (Kent) and three guests.
Now, how many women were at the dinner? Since three men arrived at the
dinner, Hannah, the co-host, kissed with each of them, resulting in
three kisses. 9 kisses are still left. If only one woman arrived, then
she had to kiss Hannah (one more kiss), and even if she was alone she
had to kiss all four men (three guests and Kent) - four more kisses.
In total it makes 5 more kisses, which is not enough since 4 kisses
are still left. Thus, one woman arrived is not enough.
Letís assume that three women arrived. Counting Hannah, that would be
four of them. That would mean the women alone exchange six kisses.
Even if all women arrived with their spouses, they each would need to
kiss three men, totaled in 12 additional kisses. 6 plus 12 equals 16
kisses in total. Thus, three women arrived is too many.
The only version left - two women arrived at the dinner. Combined with
Hannah they would exchange 3 kisses between each other. Added to
Hannahís three kisses with male guests it makes 6 kisses so far. If
both arrived women are alone, then they would kiss with each of the
four men totaling in 8 kisses. 6 kisses plus 8 kisses is 2 kisses
above 12. Thus, at least one of the arrived women was with the spouse.
Then she would kiss only three men. The other arrived women would kiss
four men. It makes 7 kisses in total. Still too many. Thus, both women
arrived with their spouses resulting in three additional kisses for
each of them with other men. In total it makes 12 kisses which finally
satisfy the condition of the puzzle.
Letís summarize. Three men arrived at the dinner. Two of them arrived
with their spouses, and one - alone. As a result 6 handshakes and 12
kisses have been done.