Problem of the Month(April 2019)

Tiling identical polykings in a rectangle have been studied, for example this page or the bottom of this page. But in order to make physical packing puzzles out of these, we need to insist that polykings not cross, such as the tiling below. What polykings (that are not polyominoes) are rigid-rectifiable in this way?

Also of interest would be using identical rigid polykings in packing puzzles with unique solutions, up to rotation and reflection, such as the packing below.

Both of these are somewhat trivial. Are there less trivial solutions?

George Sicherman confirmed that the only rectifiable non-polyomino rigid tetraking is the one above, and there are no pentakings that work.

Here are the unique packings of tetrakings and pentakings that involve at least 3 of the tile:

Tetrakings in Rectangles
n \ m23456
4 none none 3
4
5 none 3 3
4
3
4
6 none 3 3
4
5
5
6
?
7 none 3 3
4
5
6
(GS) ?
8 3 3
4
5
6
7
none ?

Pentakings in Rectangles
n \ m34567
5 none 3

3

4

6 3

3

(GS)

3
4

5
3
4

5
7 3

3

4

3
4
5

6
5
6

7

(all GS)

6
8

(all GS)

8 3

3

4

5
6
4
5
6

7

(all GS)

6
7
8

(all GS)

7
8

10

(all GS)

9 3

4
3 (GS)
4

5 (GS)
5
6
7

8

(all GS)

9
10

(all GS)

10

(all GS)

George Sicherman was interested in pairs of rigid polykings tiling rectangles:

3-kings and 3-kings

3-kings and 4-kings
4-kings and 4-kings

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/1/19.