The 1990 Game Boy game Daedalian Opus is essentially a series of 36 pentomino puzzles. In the first level, however, you only have three pentominos; the rest are introduced in the levels after that, one at a time. A good proportion of the shapes you have to fit the pentominos into are also plain rectangles. (See a tool-assisted speedrun of the game here).
That led me to the following problem. What is the smallest $k$ such that there is an ordering of the 12 distinct free pentominos where
- the first $k$ can tile a $5×k$ rectangle exactly
- the first $k+1$ can tile a $5×(k+1)$ rectangle exactly
- and so on, to the first $12$ tiling a $5×12$ rectangle exactly (which can always be done)?
For example, if $k=9$ there would have to exist an ordering – say
PINWVZXFTUYL – where
PINWVZXFTUYtiles a $5×11$ rectangle
PINWVZXFTUtiles a $5×10$ rectangle
PINWVZXFTtiles a $5×9$ rectangle