A game named Blinx, was manufactured last in 2004 by a company named Microsoft. In 2014-2015 they dropped the trademark, and so the name is usable, but the game, the intellectual material is copyrighted. If I don't use their images or their code, can I legally reproduce the game?
If you mean to "make a game with the same game rules and logic, but with new code & new graphics", then yes, you can.
To prevent someone from making a game with the exact (or very similar) rules, you'd need a patent. Two famous games which are patented are "Monopoly" (they got the name right!) and Tetris*. There is a reason why you see people making many Reversi, Sudoku, Tic-Tac-Toe etc implementations in the app store, but not Monopoly and Tetris.
*Technically, Tetris is not protected by patent, only by copyright. Many developers have rejected the company's copyright infringement claims as game design is not covered by copyright.