Ponder This Challenge - June 2026 - The Superhero Team Movies

A certain comic book publisher holds the copyright for a certain superhero team, and wishes to create a movie franchise based on them. To minimize superhero fatigue with the audience, the movie franchise is planned ahead with the goal of having the minimum number of action sequences which cover all the possible superhero combinations.

Each superhero movie is built in the same manner: First, one hero appears and participates in an action sequence. Next, another hero joins them and the next action sequence contains both heroes, and so on. A movie need not contain all the heroes on the team.

As an example, assume the superhero team contains the heroes Dinolady (D), Vacuum Cleaner (V), and MathMan (M). A possible list of movies is:

The first movie provides action sequences starring D solo, and D and V together. Mathematically speaking, {D} and {D,V}.

The second movie includes the scenes {V} and {M,V}.

The third movie has the scenes {M}, {D, M} and the grand finale of {D, M, V}.

In this example, we managed to cover all the possible superhero combinations without using any such combination twice, using a total of 7 combinations. In general this is not possible. With 4 superheroes there are 15 combinations total, but we will need at least 17 combinations spread across 6 films to cover them all (since we have 6 films and 4 superheroes, we'll have two solo action sequences starring a hero that already had a solo action sequence).

In order to write solutions compactly, we can avoid spaces and linebreaks and use "0" to signify a new movie begins. With these conventions, the above solution becomes "0DV0VM0MDV".

Your goal: Find an optimal solution for the case of $n=6$ heroes. Give your solution in the compacted format.

A bonus "*" will be given for finding an optimal solution for the case of $n=10$ heroes.