Shut the Box

"Shut the Box" is a single-player game that begins with a set of pieces numbered from $1$ to $N$. All pieces are initially unmarked (in the picture, unmarked pieces are in the upright position). In this version of the game, a player is allowed up to $T$ turns, each defined by an independently chosen value $V$ (typically determined by rolling one or more dice). During a turn, the player must select a set of currently unmarked pieces whose numeric labels sum exactly to $V$, and then mark them. The game continues either until the player has used all turns, or until a turn occurs in which it is impossible to select a set of unmarked pieces summing to the given value $V$ (in which case that turn and all remaining turns are forfeited). The goal is to mark as many pieces as possible; marking all pieces is called shutting the box. Your task is to determine the maximum number of pieces that can be marked using a fixed sequence of turn values.

As an example, consider a game with $6$ pieces and the following sequence of turn values: 10,\ 3,\ 4,\ 2. The best result for this sequence is to mark a total of four pieces. This can be achieved by using the value $10$ to mark the pieces 1 + 4 + 5, then using the value $3$ to mark piece $3$. At this point the game ends, as it is impossible to exactly use the turn with value $4$ (the final turn with value $2$ must also be forfeited). An alternative strategy to achieve the same count is to use $10$ to mark four pieces 1 + 2 + 3 + 4, with the game ending on the turn with value $3$. There is no way to mark five or more pieces with that sequence.

Hint: Avoid enormous arrays or lists, if possible.