Logical Reasoning: Puzzles, Spatial Transformations, and Game Theory

Problem Statement

You are given four identical L-shaped pieces, each covering exactly 4 unit squares in the shape of an "L" (three squares in a row with one additional square attached at one end). These pieces can be rotated (0°, 90°, 180°, 270°) but cannot be flipped or reflected. The task is to arrange these four L-shaped pieces to completely cover a 4x4 square grid, with no overlapping squares. The first piece is already placed with its corner (the junction of the L) at position (2,2) on the grid, and its long arm extending horizontally to the right, while the short arm extends vertically upward. The grid has a total of 16 unit squares, and all must be covered exactly once. In how many different ways can the remaining three L-shaped pieces be arranged to complete the 4x4 square? 1 2 3 4 A 3D L-shaped block (3x2x1 units with a 1x1x1 cube at its corner) has one red face (originally front), one blue face (originally top), and other faces are white. It undergoes the following sequence: 1) Rotate 90° cloc