Can you tile the remaining 24-unit shape with 1×3 "trominoes" (three squares in a straight line)?

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.

Now remove the top-left corner (1,1). Its color is (1+1) mod 3 = 2 mod 3 = Color 2? Wait — careful: (1+1)=2, so 2 mod 3 = 2 — yes, Color 2. So after removal: Color 0: 9 Color 1: 8 Color 2: 7 (since we removed one from Color 2) Each 1×3 tromino, no matter how you place it (horizontal or vertical), covers exactly one square of each color .

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 our shape after removing a corner has: Color 0: 9 Color 1: 8 Color 2: 7

We have (9 instead of 8) and too few Color 2 (7 instead of 8). Impossible. 6. The Beautiful Conclusion The tiling fails not because of a bad arrangement, but because of an invariant — a numerical property preserved by every tromino but violated by the board’s initial coloring counts.

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 .