Thus, . 5. The Contradiction If 8 trominoes tile the shape, they would cover: 8 trominoes × 1 square of each color = 8 of Color 0, 8 of Color 1, 8 of Color 2.
But ? 2. The First Attempt You try. Place a tromino horizontally in the top row. Then another. You quickly get stuck — the missing corner leaves an awkward gap. After some attempts, you suspect it’s impossible . elementary mathematics dorofeev
Try to visualize: the 5×5 board has 25 squares. Remove one corner → 24 squares. Each tromino covers 3 squares. 24 ÷ 3 = 8 trominoes needed. So numerically it’s possible. Place a tromino horizontally in the top row
Proof: Horizontal tromino covers cells (r,c), (r,c+1), (r,c+2). Their (row+col) mod 3 = (r+c) mod 3, (r+c+1) mod 3, (r+c+2) mod 3 → three consecutive integers mod 3 → all different residues 0,1,2. Same for vertical. but in three colors repeating diagonally:
Here’s an original, interesting piece inspired by the style and depth of Elementary Mathematics by Dorofeev (known for its elegant problems, surprising connections, and geometric intuition). The Square That Didn't Want to Be Alone A Dorofeev-style exploration: How a simple geometric puzzle hides a deep number theory secret. 1. The Puzzle (seems easy, but wait...) Take a 5×5 square made of 25 unit squares. Remove one corner unit square.
Can you tile the remaining 24-unit shape with 1×3 "trominoes" (three squares in a straight line)?
Why? Color the 5×5 board in a clever way — not like a chessboard (alternating black-white), but in three colors repeating diagonally: