Problem of the Month (May 2007)

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
40

Heptominoes

2
4
7
8
10
12
16
18
20
24
26
32
36
40
44
64
84

The Largest Octominoes

48
52
56
60
76
80

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: . The unsolved cases for n=8 are shown in shown below.

In 2018, George Sicherman considered one-sided polyominoes, and found these solutions that differed from the two-sided case:

One-Sided, Row and Column

Pentomino 24

Hexomino 24

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