431. From $a$ cards, $b$ piles are formed, with each pile having $c$ points. The first card of each pile counts as 11 points if it is an ace. 10 points if it is a face card or a ten, and so on. Other cards count as one point. If, after forming the piles, we have $d$ cards left over, we need to determine the sum $\boldsymbol{x}$ of the points of the first cards only.
431. \( b c \) is the total sum of points of all piles. If you subtract the cards that count as one point, the remainder will be \( x \), but the number of these cards is \( a-d \) without the number of cards counted as more than one point, i.e., \( a-d-b \). Therefore:

\[
x = b c - (a - d - b) = b(c + 1) + d - a
\]