Everybody was found to have kissed everybody else once under the mistletoe, with the following additions and exceptions: 
No male kissed a male; no man kissed a married woman except his own wife; 
all the bachelors and boys kissed all the maidens and girls twice; 
the widower did not kiss anybody, and the widows did not kiss each other. 
Every kiss was returned, and the double performance was to count as one kiss. 
In making a list of the company, we can leave out the widower altogether, because he took no part in the osculatory exercise.

Now, if every one of these 39 persons kissed everybody else once, the number of kisses would be 741; 
and if the 12 bachelors and boys each kissed the 10 maidens and girls once again, we must add 120, making a total of 861 kisses.

But as no married man kissed a married woman other than his own wife, we must deduct 42 kisses; as no male kissed another male, we must deduct 171 kisses; 
and as no widow kissed another widow, we must deduct 3 kisses. 
We have, therefore, to deduct 42+171+3=216 kisses from the above total of 861, and the result, 645, represents exactly the number of kisses that were actually given under the mistletoe bough.