Ken's Puzzle of the Week

Dividing A Square

Dividing along the gridlines, divide an NxN square in to the maximum number of pieces, such that each piece is unique (including rotations and reflections.)  Solve for N<=8.  For example, for N=3, you can divide a 3x3 square into four pieces of sizes (3,3,2,1).

Source: Original.

Solutions were received from David Madfes, Alan O'Donnell, Bojan Basic, Joseph DeVincentis, Claudio Baiocchi, Dan Chirica, Bernie Erickson, and Yaacov Yoseph Weiss.  It was found that for each square, the maximum theoretical number of pieces was achievable.

[Update 24Sept2008: Submitted this sequence to the OEIS:
http://www.research.att.com/~njas/sequences/A144876]

Bernie Erickson has this summary:

Shapes can be formed:  1 one, 1 two, 2 threes, 5 fours, and 12 fives.

The series then:  1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, . . .

The running sum:  1, 3, 6, 9, 13, 17, 21, 25, 29, 34, 35, 39, 44, 49, 54, 59, 64, . . .

Square numbers in the series fit inside:  1, 6, 9, 17, 25, 39, 49, 64, . . .

Maximal candidates are:  1, 6 - 2, 9, 17 - 1, 25, 39 - 3, 49, 64

Note that with three of the eight N x N squares, a larger piece is needed but can be offset by just one smaller piece taken away.

Candidates

1 Piece,         1 x 1:  1

2 Pieces,       2 x 2:  1, 3

4 Pieces,       3 x 3:  1, 2, 3, 3

5 Pieces,       4 x 4:  2, 3, 3, 4, 4

8 Pieces,       5 x 5:  1, 2, 3, 3, 4, 4, 4, 4

10 Pieces,     6 x 6:  1, 2, 3, 4, 4, 4, 4, 4, 5, 5

13 Pieces,     7 x 7:  1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5

16 Pieces,     8 x 8:  1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5

 

All the candidates check out.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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