Problem #52 MEDIUM
The Mango Orchard Division
Scenario Geometry Combinatorics Logic
Problem Statement
In a small village near Ratnagiri, old Ramabai has four grown children — Sunita, Mohan, Priya, and Dev — and she has decided it is time to divide her mango orchard among them before she gets any older.
The orchard is a perfect square plot, 100 metres on each side. Ramabai has one condition that she will not bend on: each child must receive a plot of exactly equal area. She also wants each child's plot to be a single connected piece of land — no child should end up with two separate patches.
Sunita, who studied geometry in college, starts sketching possible divisions in her notebook. Mohan says it is impossible to divide a square into four equal connected pieces that are not simply four smaller squares. Priya laughs and says she can think of at least five different ways to do it.
Dev pulls out a piece of paper and asks: 'How many geometrically distinct ways can we divide a 4x4 grid into four congruent connected pieces?'
How many distinct ways can you divide a 4x4 grid into exactly four congruent (same shape and size) connected pieces, each containing exactly 4 cells?
Answer & Quick Explanation
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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.
Detailed Editorial Solution
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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.