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Checkerboard Dissection (Posted on 2009-11-23) Difficulty: 2 of 5
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.

See The Solution Submitted by Brian Smith    
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Solution Solutions and Explorations | Comment 2 of 3 |

The requirements are met here along with a few other thoughts.<o:p></o:p>

Each division is represented in each solution with a different format.<o:p></o:p>

<o:p></o:p>

Uses 4 divisions.
Uses a 3 x 3  division and 1 rotation.
  1   2   3   4    5          1   2   3
  6   7   8   9  10          6   7   8        5  10  15  16

11 12  13 14  15        11  12 13        4     9  14  21
16 17  18  19  20                           17  18  19  20
21 22  23  24  25                           22  23  24  25<o:p></o:p>

 <o:p></o:p>

Uses 4 divisions.
Uses a 3 x 3 block as one entity but has no rotations or reflections; all shapes have different areas.
 1   2   3   4    5          1    2    3
 6   7   8   10          6    7    8         16   17    4     5

11 12 13  14 15        11   12  13         21   22    9   10
16 17 18  19 20                                18   19   14   15
21 22 23  24 25                                23   24   25   20<o:p></o:p>

<o:p></o:p>

Uses 4 divisions.
Has no 3x3 or 4x4 division but uses 1 rotation
 1    2    3       5         1    2    3
 6    7    8    9  10         6    7    8           21  16   9  10

11  12  13   14  15       11   4     5           12  13  14  15
16  17  18   19  20                               17  18  19  20
21  22  23   24  25                               22  23  24  25<o:p></o:p>

<o:p></o:p>

Uses 4 divisions
Has a 4x4 division and 2 rotations
  1   2   3   4     5          1     2    3
  6   7   8   9   10         6    21   4         7     8   9  10

11  12 13  14  15        11    16    5       12   13  14  15
16 17  18  19  20                               17  18   19  20
21 22  23  24  25                               22  23   24  25

<o:p></o:p>

Uses 5 divisions, 4 make up the 3 x 3.
Uses a 4 x 4 block as one entity.  No shape is rotated or reflected.
  1   2   3    4     5         1    2   11
    7   8   9   10         6  21   16         7    8    9  10   

11  12 13  14   15        3     4      5        12  13  14  15
16  17 18  19   20                              17  18  19  20
21  22 23  24   25                              22  23  24  25<o:p></o:p>

 <o:p></o:p>

Uses 5 divisions of which none is a 3 x 3 or a 4 x 4.  In forming those squares there has been no rotation or reflection.
  1   2   3    4    5          1    2   11
 6    7   8    9  10         6    7   16        21   8   9   10

11  12  13  14  15        3     4      5       12  13  14   15
16  17  18  19  20                             17  18  19   20
21  22  23  24  25                             22  23  24   25<o:p></o:p>

 <o:p></o:p>

Again this uses 5 divisions of which none is a 3 x 3 or a 4 x 4.  In forming those squares there has been no rotation or reflection.
 1   2    3   4   5       15   4    5             2     3   8    9
 6   7    8   9  10      20   1   10           11  12  13  14
11 12  13  14  15     25    6     7              17  18  19  20
16 17  18  19  20                               22  23  24  25
21 22  23  24  25<o:p></o:p>

<o:p></o:p>

Uses 5 divisions.
Has no 3x3 or 4x4 division but has one rotation
 1    2   3   4    5         1   2    3
 6    7     9  10          6   7    8          21   4    9   10

11  12 13  14  15        11  12   5          16   13  14   15
16  17 18  19  20                               17   18  19   20
21  22  23  24 25                               22   23  24   25


  Posted by brianjn on 2009-11-23 19:24:31
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