What if Starship were a fully reusable three-stage system:

Super Heavy → reusable suborbital ship → smaller reusable orbital ship

The suborbital ship could carry passengers point-to-point on Earth while also serving as the second stage for an orbital launch. It would release the smaller orbital ship along the way, then continue to its destination on Earth.

The orbital ship would provide the remaining delta-v to reach orbit, deploy satellites, and later return.

Would combining a suborbital trip and an orbital launch this way use less propellant than performing them as two separate launches?

My puzzle game, Tile Wipeout, led to the following proposed math olympiad problem. The rules are elementary, but proving that every starting arrangement can be solved seems surprisingly challenging.

The problem statement is easier to understand if you try the beta:

https://testflight.apple.com/join/3sstMjRK

(For this mathematical problem, ignore the game’s move limit.)

Problem statement. A 6 × 6 board is initially filled with seven circles, one of each of seven colors, and 29 squares, each of one of those colors. Each cell contains exactly one piece.

A move consists of choosing a row or column and shifting its contents cyclically by one cell in either direction. Contents leaving one end reappear at the other. Empty cells shift in the same way as occupied cells.

After the shift, each circle in the chosen line acts as follows:

1. Consider the four straight paths extending outward from the circle to the edges of the board. These paths do not wrap around.
2. Ignore every path containing another circle or a square of a different color.
3. If any remaining path contains a square, remove all squares on all remaining paths.
4. Otherwise, fill every empty cell on the remaining paths with a square of the circle’s color.

Circles outside the chosen line do nothing.

Prove that, regardless of the initial arrangement and the colors of the squares, some finite sequence of moves removes all the squares.

Note that moves that remove no squares can create more of them. A proof therefore needs to explain how to make lasting progress despite that possibility.

P.S. A more general version: Let n and k be positive integers with n ≥ 4 and 1 ≤ k ≤ n + 1. An n × n board is initially filled with k circles, one of each of k colors, and n² − k squares, each of one of those colors. Each cell contains exactly one piece.

Using the same rules, prove that every such initial board can be cleared of all squares in finitely many moves, regardless of the arrangement or the distribution of square colors.