My favorite type of physical puzzles are the Rubik-type puzzles called Magic Polyhedrons. These puzzles screams: “group theory” and “combinatorics”!
I bought about 7 varieties from mefferts puzzles in 2001.
A magic polyhedron type of puzzle is essentially defined by the cuts and the solid it cuts into. The cuts defines the mathematical essence of the puzzle, while the solid shape gives the puzzle a flavor. Same cuts can be applied to different solids, yet still resulting the same puzzle mathematically. For example, a tic-tac-toe cut applied to a cube results the Rubic's Cube, and appling it to a sphere results a Rubic Ball. The same tic-tac-toe cut can be applied to octahedron or other shapes.
Modern ones are stickerless. This is a improvement, because you don't have to worry about stickers falling out or dealing with the glue leaking on the edge of the stickers.
For software versions of magic polyhedrons, see: Great Math Software on Puzzles.
There are lots books on Rubik cube. Two “oldish” books I've read in early 1990s are Handbook of Cubik Math by D Singmaster et al. Buy at amazon. And Rubik's Cubic Compendium by E Rubik et al. Buy at amazon. I liked the first one better. Both are fairly popular.