Can you make them fit?
Fill a 7 × 7 board with seven rectangles. Each must have perimeter 12.
The clock starts when you reveal the puzzle.7 × 5 = 35
If all seven pieces were 1 × 5, they would cover only 35 of the board’s 49 cells. We need 14 more.
THE SPACE INSIDE A BOUNDARY
Same perimeter.
Different shapes.
Seven rectangles.
One shared boundary length.
Tap two corner cells, or drag to draw a rectangle.
Tap two corner cells, or drag. Tap a placed piece to remove it.AT YOUR OWN PACE
A brain snack, with room to think.
THE ANSWER OPENS ANOTHER DOOR
Now for the lovely part.
You have an answer. Explore how it works, then take the idea somewhere new.
See why it works ↓Keep your time
THE IDEA BEHIND YOUR ANSWER
One boundary length.
Three kinds of space.
Your rectangles all measure 12 units around their edges. Inside, they tell a different story. Explore the pieces in your own arrangement.
FOLLOW THE BOUNDARY
Perimeter counts the distance around a shape. Area counts the space inside it. Equal perimeters can enclose different areas.
TRY CHANGING THE SHAPE
Keep the perimeter.
Change the area.
A rectangle has perimeter 2 × (width + height). For a perimeter of 12, width + height must be 6.
Slide from long and narrow to square, then back again. Turning a rectangle around changes its orientation, not its shape.
WHY YOUR MIX OF PIECES WAS INEVITABLE
Let the area do some thinking.
With whole-number sides adding to 6, there are only three shapes up to rotation: 1 × 5, 2 × 4 and 3 × 3. Their areas are 5, 8 and 9.
+3 or +4 cells
Replacing a 1 × 5 with a 2 × 4 adds 3 cells. Replacing it with a 3 × 3 adds 4 cells.
2 × 3 + 2 × 4 = 14
The only whole-number solution to 3b + 4c = 14 is b = 2, c = 2. That leaves three of the smallest pieces.
So every valid board has three 1 × 5 rectangles, two 2 × 4 rectangles and two 3 × 3 squares. Here is the count from your arrangement:
| Shape | Count | Perimeter | Area each | Cells covered |
|---|---|---|---|---|
| Total | 7 | 12 each | — | 49 |
Area fixes the mix. Your drawing shows that the pieces can actually fit. Matching the total area alone would not prove that a tiling exists.
The idea to take away
When a packing puzzle feels like guesswork, count the area before arranging the pieces. One measurement can reveal what another leaves open.