This game is called Daughter in the Box ( 箱入り娘 hakoiri musume), and it’s based on Klotski, a game in which a specific block has to be moved to some predefined location. The player is not allowed to remove blocks, and may only slide blocks horizontally and vertically to solve the puzzle with a minimum number of moves in a minimum amount of time.
That expression, hakoiri musume, also refers to an overprotected girl who grows up in either a very wealthy or very respectable family. She’s treated like a treasure that may never be removed from its package, so that she can be presented in a perfect condition to her groom on her wedding day.
There are surprisingly (or perhaps not?) many hakoiri musume in Japan.
The game was a gift from a friend who often visits Takayama, a city famous for its trees and wood. The different blocks are made from different types of wood.