A 5x5 square has a checkerboard pattern. I want to divide the square into pieces to form a 4x4 and a 3x3 square. Both smaller squares will retain the checkerboard pattern.

Find a four piece division of the 5x5 square that will accomplish this.

Find a five piece division which accomplishes the task without needing to flip or rotate any of the pieces.

Part 1:

+---+---+---+---+---+
|   |   |   |   | x |
|   |   |   |   | 3 |
+---+---+---+---+---+
|   |   |   |   |   |
|   |   |   |   | 3 |
+---+---+---+---+---+
|   |   |   |   | x |
|   |   |   |   | 2 |
+---+---+---+---+---+
|   |   |   |   |   |
|   |   |   |   | 2 |
+---+---+---+---+---+
| x |   | x |   | x |
| 1 | 1 | 2 | 2 | 2 |
+---+---+---+---+---+

The 4x4 board is left intact and has been shown as blank squares.

The pieces making up the 3x3 board have been numbered 1, 2 and 3, with dark squares marked with an x. Piece 2 forms a framework into which pieces 1 and 3, each containing a dark and a light square can be placed. Piece 1 need not be rotated so long as it goes into the top row of the 3x3 frame; then piece 3 is rotated 90° clockwise to go into the middle row.

Part 2:

+---+---+---+---+---+
|   |   |   |   | x |
|   |   |   | 3 | 3 |
+---+---+---+---+---+
|   |   |   | x |   |
|   |   |   | 3 | 3 |
+---+---+---+---+---+
|   |   |   |   | x |
|   |   |   | 2 | 2 |
+---+---+---+---+---+
|   | x |   | x |   |
| 1 | 1 | 1 | 2 | 2 |
+---+---+---+---+---+
| x |   | x |   | x |
| 1 | 1 | 1 | 1 | 2 |
+---+---+---+---+---+

The top-left 3x3 is left as-is.

Piece 3 is placed above the left two columns of piece 1.

Piece 2 is ratcheted up and to the left, completing the 4x4.

This has been done with 4 pieces, with none flipped or rotated. If you really need 5 pieces, just cut one apart :-).

 

Posted by Charlie on 2009-11-23 15:55:00