The Mango Orchard Division

A 4x4 grid can be divided into 4 congruent connected pieces in several distinct ways. The I-tetromino allows 2 tilings (all horizontal or all vertical). The O-tetromino (2x2 square) allows 1 tiling. The L-tetromino allows pinwheel-style tilings. The T and S/Z tetrominoes cannot tile the grid with 4 identical copies.

A 4x4 grid has 16 cells. Dividing it into 4 congruent connected pieces means each piece has 4 cells — these are the 5 types of tetrominoes. Test each tetromino type for whether 4 copies can tile the 4x4 grid. Step 1: I-tetromino (straight line of 4): Place all 4 horizontally — tiles rows 1, 2, 3, 4 perfectly. Place all 4 vertically — tiles columns 1, 2, 3, 4 perfectly. Two distinct tilings. Step 2: O-tetromino (2x2 square): Four 2x2 squares fit exactly into the 4x4 grid (top-left, top-right, bottom-left, bottom-right). One distinct tiling. Step 3: T-tetromino: Attempting to tile 4x4 with 4 T-pieces fails — a checkerboard colouring argument shows T-pieces always cover 3 cells of one colour and 1 of another, making equal coverage of both colours impossible for 4 copies in a balanced grid. Step 4: S/Z-tetromino: These skewed pieces cannot tile a 4x4 grid with 4 identical copies without gaps or overlaps — they always leave uncoverable corners. Step 5: L/J-tetromino: 4 L-pieces can tile the 4x4 grid in several orientations — for example, two L-pieces pointing one way and two the other in a pinwheel arrangement. Step 6: The total number of ways to partition a 4x4 into 4 congruent connected tetromino-shaped pieces is generally given as 4 canonical families. Key Insight: Tetromino tiling problems are a beautiful intersection of combinatorics and geometry. The checkerboard colouring argument is the most powerful tool: colour the grid alternately black and white, then count how many cells of each colour each piece covers. If the counts are incompatible with balanced coverage, that piece cannot tile the grid.