Pairwise Touching Polyominoes In Boxes

We say a collection of polyominoes is pairwise touching if every polyomino touches every other along at least one face. If we cut up an n x m x p box into polyominoes, what is the maximum number of pairwise touching polyominoes we can get? The tables below show the best-known bounds. Click on the links for pictures.

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Pairwise Touching Polyominoes in n x m x 1 Box

n \ m	1	2	3	4	5
1	1	2	2	2	2
2	2	3	3	3	3
3	2	3	4	4	4
4	2	3	4	4	4
5	2	3	4	4	4

Pairwise Touching Polyominoes in n x m x 2 Box

n \ m	1	2	3	4	5
1	2	3	3	3	3
2	3	4	4	4	4
3	3	4	5	6	6
4	3	4	6	7	7,8
5	3	4	6	7,8	7,10

Pairwise Touching Polyominoes in n x m x 3 Box

n \ m	1	2	3	4	5
1	2	3	4	4	4
2	3	4	5	6	6
3	4	5	5	6	7
4	4	6	6	8	8,10
5	4	6	7	8,10	9,13

Pairwise Touching Polyominoes in n x m x 4 Box

n \ m	1	2	3	4	5
1	2	3	4	4	4
2	3	4	6	7	7,8
3	4	6	6	8	8,12
4	4	7	8	9,12	9,16
5	4	7,8	8,12	9,16	10,18