6. (10 points) A deck of playing cards, excluding the joker, has 4 suits totaling 52 cards, with each suit having 13 cards, numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of these 14 cards Feifei drew is exactly 35, then how many of them are 1? 

$\qquad$ cards are 1.
【Solution】Solution: According to the analysis, the minimum sum of the two hearts is $1+2=3$, the minimum sum of the three spades is $1+2+3=6$,

the minimum sum of the four diamonds is $1+2+3+4=10$, and the sum of the five clubs is: $1+2+3+4+5=15$, so the minimum sum of the 14 cards is: $3+6+10+15=34$,
(1) If only $1 \sim 2$ cards are 1, the sum of the card faces is clearly greater than 35;
(2) If three are 1, then the minimum sum of the card faces is: $2+3+1+2+3+1+2+3+4+1+2+3+4+5=36>35$, which contradicts the problem statement;
(3) If four are 1, then the minimum sum of the card faces is: $3+6+10+15=34<35$, which meets the requirement.

Therefore: There are four 1s.
The answer is: 4.