L Shapes from Pentacube Pairs


A polycube is a solid made of equal cubes joined face to face, and a pentacube is a polycube with 5 cells. There are 29 pentacubes, distinguishing mirror images:

The letters shown in black are my names. Mirror images of chiral pentacubes are indicated with ; e.g., G′ is the mirror image of G. The green symbols are Kate Jones's names. The red symbols are Donald Knuth's names.

I define an L-shaped polycube as a polycube prism whose base is L-shaped; that is, it consists of a rectangle from one corner of which a smaller rectangle has been excised.

Here I show the smallest known L-shaped polycubes that can be tiled with a given pair of pentacubes, using at least one of each. Chiral pairs of pentacubes are distinguished, and chiral pentacubes may not be reflected when used in these tilings.

If you find a smaller solution, please write.

See also L Shapes from Two Pentominoes.

Table of Results

A 661214613666364268126888067
B  31218372961965351212381420614
E   8912962566663632384646691666
F    1897495128349412241542512
G      ×1282126126567635630121821631844087
H        456664664286641064614610
I       8122116241147627857
J           3627633876334491244
K         88634291068612612
L          8224646323757
M           105619915315920810
N            246766761466
P             2532224622
Q              423234644
R                   301261268918716
S                     4156342484
T                 312972821
U                  612326
V                   92482
W                    28616
X                     10126
Y                      5


The pair G and G′ cannot tile any L-shaped polycube. Each of the solutions shown for pairs A-X, C-X, and X-Z was formed by joining two rectangular boxes. Smaller solutions may well exist for these.

2 Tiles

3 Tiles

4 Tiles

5 Tiles

6 Tiles

7 Tiles

8 Tiles

9 Tiles

10 Tiles

11 Tiles

12 Tiles

13 Tiles

14 Tiles

15 Tiles

16 Tiles

18 Tiles

19 Tiles

20 Tiles

21 Tiles

24 Tiles

28 Tiles

30 Tiles

42 Tiles

56 Tiles

72 Tiles

80 Tiles

126 Tiles

440 Tiles

Last revised 2022-10-31.

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Col. George Sicherman [ HOME | MAIL ]