What were those old puzzle games with the movable squares called?

The puzzle games with the movable squares you're thinking of are most likely called "sliding tile puzzles" or "15-puzzles".

Here's why:

* Sliding Tile Puzzles: This is the most common and general term for these games, as it describes the core gameplay mechanic.

* 15-Puzzle: This refers to a specific version of the sliding tile puzzle with 15 numbered tiles and one empty space. It's one of the most well-known variations.

You might also encounter other names like:

* "Number Puzzle"

* "Fifteen Puzzle" (variant spelling)

* "Gem Puzzle" (when the tiles have different images instead of numbers)

Regardless of the name, the basic principle remains the same: sliding the tiles around to arrange them in a specific order, usually by filling the empty space with the missing tile.