Pick a polyomino. Arrange as few copies as possible in a square so that the same number of squares in every row and column are covered.
What are the best results for small polyominoes? hexominoes? heptominoes? What patterns can you find? Which polyominoes are impossible? If we relax the requirements, what is the smallest rectangle containing copies of a polyomino so that the same number of squares in each column are covered? What are the solutions for polyhexes or polyiamonds or other polyforms?
ANSWERS
To pack m copies of an n-omino into a square of side s with c squares covered in each row and column, Livio Zucca gave the trivial necessary condition n m = s c.
The orange figures were found by me. The green figures were found by George Sicherman.
Polyominoes, Row and Column
Small Polyominoes
Tetrominoes
1
4
Pentominoes
2
4
5
8
12
24
Hexominoes
2
4
6
8
10
12
18
24
36
56
Heptominoes
2
4
7
8
10
12
16
18
20
24
26
32
36
40
44
64
84
The missing n-ominoes are impossible for n≤6. Joseph DeVincentis and George Sicherman proved the Z pentomino was impossible. There are 5 unsolved cases for n=7:
Polyominoes, Column
Small Polyominoes
Pentominoes
1
2
3
6
Hexominoes
1
2
4
6
George Sicherman proved that if a polyomino has an orientation where the first few columns all contain a squares, and the rest of the columns all contain b squares, the polyomino has a one-way solution. He conjectures that if a polyomino has a monotone shadow in some direction, the polyomino has a one-way solution.
Polyiamonds, Row and One Column
Small Polyiamonds
Pentiamonds
2
4
Hexiamonds
1
2
3
4
6
Heptiamonds
2
4
5
6
8
Polyiamonds, One Column
Small Polyiamonds
Hexiamonds
1
2
Heptiamonds
1
2
3
6
Polyhexes, Row and One Column
Small Polyhexes
Tetrahexes
1
2
3
Pentahexes
1
2
3
4
6
8
12
15
Hexahexes
1
2
3
4
6
8
9
10
12
14
20
54
The missing polyhexes are impossible.
Polyhexes, One Column
Small Polyhexes
Pentahexes
1
2
3
Polykings, Row and Column
Small Polykings
Tetrakings
1
2
4
6
If you can extend any of these results, please
e-mail me.
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