A 4x4x2 solid contains 32 cubes. This can be filled with three copies each of
two pentacubes, with two cubes left unfilled. Here we require that the two
missing cubes be adjacent, or, equivalently, that we add a dicube to the set.
Below is a catalog of the pentacube pairs for which solutions exist. These
are qualified by the placement of the dicube, which can be in one of ten
placements, arbitrarily labeled here as A, B, C, D, E, F, G, H, J, or K. For
example, the first set shown below, the combination of
three F- and three Q-pentacubes, has only one
solution, and that has the dicube placed at A. Similarly,
three P- and three T-pentacubes can only be solved
with the dicube at D.
Many sets can be solved with the dicube at different places. For example,
three P- and three N1-pentacubes can be solved
with the dicube nearly anywhere, but for most placements of the dicube the
solution is unique.
The catalog links to solution drawings whenever there are three or fewer for a
particular dicube placement. For example, to see the two solutions for
three Y- and three T2-pentacubes, click the
solution drawings link. In the bottom half of
the linked window you will see a frame for each solution. Within the frame
you can click on the next (or previous) link to add (or remove) a piece from
the drawing.
1: 3 F-pentacubes, 3 Q-pentacubes
1 solution for A
3× 
3× 
solution drawings
2: 3 L3-pentacubes, 3 N2-pentacubes
1 solution for A
3× 
3× 
solution drawings
3: 3 L2-pentacubes, 3 L3-pentacubes
1 solution for A
3× 
3×
solution drawings
4: 3 L4-pentacubes, 3 V2-pentacubes
1 solution for A
3× 
3× 
solution drawings
5: 3 T-pentacubes, 3 Q-pentacubes
1 solution for B
3× 
3×
solution drawings
6: 3 P-pentacubes, 3 T-pentacubes
1 solution for D
3× 
3×
solution drawings
7: 3 S-pentacubes, 3 L2-pentacubes
1 solution for D
3× 
3×
solution drawings
8: 3 U-pentacubes, 3 L3-pentacubes
1 solution for D
3× 
3×
solution drawings
9: 3 Q-pentacubes, 3 T1-pentacubes
1 solution for D
3×
3× 
solution drawings
10: 3 L1-pentacubes, 3 N1-pentacubes
1 solution for D
3× 
3× 
solution drawings
11: 3 L4-pentacubes, 3 T1-pentacubes
1 solution for D
3×
3×
solution drawings
12: 3 Y-pentacubes, 3 A-pentacubes
1 solution for E
3× 
3× 
solution drawings
13: 3 L4-pentacubes, 3 N1-pentacubes
1 solution for E
3×
3×
solution drawings
14: 3 U-pentacubes, 3 N1-pentacubes
1 solution for F
3×
3×
solution drawings
15: 3 U-pentacubes, 3 L4-pentacubes
1 solution for J
3×
3×
solution drawings
16: 3 V-pentacubes, 3 L4-pentacubes
1 solution for J
3× 
3×
solution drawings
17: 3 L4-pentacubes, 3 T2-pentacubes
1 solution for A and 1 solution for C
3×
3× 
solution drawings
18: 3 Y-pentacubes, 3 T2-pentacubes
1 solution for D and 1 solution for E
3×
3×
solution drawings
19: 3 L4-pentacubes, 3 S1-pentacubes
1 solution for A and 1 solution for F
3×
3× 
solution drawings
20: 3 L-pentacubes, 3 L1-pentacubes
1 solution for C and 1 solution for G
3× 
3×
solution drawings
21: 3 Y-pentacubes, 3 L4-pentacubes
1 solution for D and 1 solution for G
3×
3×
solution drawings
22: 3 Y-pentacubes, 3 N2-pentacubes
1 solution for F and 1 solution for J
3×
3×
solution drawings
23: 3 Y-pentacubes, 3 N1-pentacubes
1 solution for E, 1 solution for F and 1 solution for J
3×
3×
solution drawings
24: 3 L-pentacubes, 3 N1-pentacubes
1 solution for B, 1 solution for D, 1 solution for G and 1 solution for H
3×
3×
solution drawings
25: 3 L3-pentacubes, 3 L4-pentacubes
1 solution for D, 1 solution for E, 1 solution for F and 1 solution for J
3×
3×
solution drawings
26: 3 L2-pentacubes, 3 S1-pentacubes
1 solution for A, 1 solution for B, 1 solution for D, 1 solution for H and 1 solution for J
3×
3×
solution drawings
27: 3 P-pentacubes, 3 N1-pentacubes
1 solution for A, 1 solution for B, 1 solution for E, 1 solution for F, 1 solution for H, 2 solutions for G, 2 solutions for J, 2 solutions for K and 4 solutions for D
3×
3×
solution drawings
28: 3 L-pentacubes, 3 A-pentacubes
1 solution for E, 1 solution for F, 1 solution for G and 2 solutions for D
3×
3×
solution drawings
29: 3 S-pentacubes, 3 L4-pentacubes
1 solution for C, 1 solution for E, 1 solution for J and 2 solutions for D
3×
3×
solution drawings
30: 3 Q-pentacubes, 3 T2-pentacubes
1 solution for D, 1 solution for E and 2 solutions for G
3×
3×
solution drawings
31: 3 L-pentacubes, 3 L2-pentacubes
1 solution for B, 1 solution for H, 2 solutions for E and 2 solutions for J
3×
3×
solution drawings
32: 3 P-pentacubes, 3 L1-pentacubes
1 solution for A, 1 solution for J, 2 solutions for E, 2 solutions for F and 5 solutions for D
3×
3×
solution drawings
33: 3 P-pentacubes, 3 Y-pentacubes
1 solution for B, 1 solution for C, 2 solutions for A, 3 solutions for G, 5 solutions for F, 8 solutions for E and many solutions for D
3×
3×
solution drawings
34: 3 L1-pentacubes, 3 Q-pentacubes
1 solution for D, 1 solution for G, 4 solutions for A and 7 solutions for E
3×
3×
solution drawings
35: 3 S-pentacubes, 3 L3-pentacubes
1 solution for B and 2 solutions for A
3×
3×
solution drawings
36: 3 L-pentacubes, 3 T2-pentacubes
1 solution for F and 2 solutions for D
3×
3×
solution drawings
37: 3 S-pentacubes, 3 T2-pentacubes
1 solution for F and 2 solutions for D
3×
3×
solution drawings
38: 3 U-pentacubes, 3 Y-pentacubes
1 solution for B and 2 solutions for F
3×
3×
solution drawings
39: 3 L-pentacubes, 3 N2-pentacubes
1 solution for J and 2 solutions for E
3×
3×
solution drawings
40: 3 L-pentacubes, 3 U-pentacubes
1 solution for F, 2 solutions for A, 2 solutions for D and 3 solutions for E
3×
3×
solution drawings
41: 3 P-pentacubes, 3 A-pentacubes
1 solution for B, 2 solutions for E, 2 solutions for G, 3 solutions for F and many solutions for D
3×
3×
solution drawings
42: 3 L-pentacubes, 3 L4-pentacubes
1 solution for K, 2 solutions for F, 3 solutions for B, 3 solutions for C, 3 solutions for E, 3 solutions for G, 3 solutions for H, 4 solutions for A, 6 solutions for D and 7 solutions for J
3×
3×
solution drawings
43: 3 L-pentacubes, 3 Y-pentacubes
1 solution for F, 2 solutions for B, 3 solutions for D and 4 solutions for E
3×
3×
solution drawings
44: 3 L-pentacubes, 3 S-pentacubes
1 solution for E, 2 solutions for B, 8 solutions for F and many solutions for D
3×
3×
solution drawings
45: 3 P-pentacubes, 3 L3-pentacubes
1 solution for A, 2 solutions for B, many solutions for D, many solutions for E, many solutions for F and many solutions for G
3×
3×
solution drawings
46: 3 S-pentacubes, 3 Y-pentacubes
1 solution for A, 3 solutions for C, 3 solutions for F and 3 solutions for G
3×
3×
solution drawings
47: 3 L3-pentacubes, 3 Q-pentacubes
1 solution for C, 3 solutions for A, 4 solutions for B, 4 solutions for D, 7 solutions for F and many solutions for E
3×
3×
solution drawings
48: 3 L2-pentacubes, 3 Q-pentacubes
1 solution for C, 4 solutions for E, 6 solutions for F, 6 solutions for K, 7 solutions for A, 8 solutions for B, 8 solutions for H, many solutions for D, many solutions for G and many solutions for J
3×
3×
solution drawings
49: 3 N1-pentacubes, 3 Q-pentacubes
1 solution for C, 5 solutions for F, 5 solutions for J, 6 solutions for B, 6 solutions for H, 7 solutions for A, 7 solutions for E, 7 solutions for G, 7 solutions for K and many solutions for D
3×
3×
solution drawings
50: 3 L4-pentacubes, 3 Q-pentacubes
1 solution for G, 5 solutions for K, 7 solutions for B, 7 solutions for H, many solutions for A, many solutions for D, many solutions for E, many solutions for F and many solutions for J
3×
3×
solution drawings
51: 3 S-pentacubes, 3 Q-pentacubes
1 solution for E, 7 solutions for G, many solutions for A and many solutions for D
3×
3×
solution drawings
52: 3 U-pentacubes, 3 N2-pentacubes
2 solutions for A
3×
3×
solution drawings
53: 3 N2-pentacubes, 3 Q-pentacubes
2 solutions for A
3×
3×
solution drawings
54: 3 V-pentacubes, 3 Q-pentacubes
2 solutions for D
3×
3×
solution drawings
55: 3 Q-pentacubes, 3 V1-pentacubes
2 solutions for E
3×
3× 
solution drawings
56: 3 L4-pentacubes, 3 S2-pentacubes
2 solutions for G
3×
3× 
solution drawings
57: 3 A-pentacubes, 3 L2-pentacubes
2 solutions for J
3×
3×
solution drawings
58: 3 L2-pentacubes, 3 L4-pentacubes
2 solutions for A and 2 solutions for J
3×
3×
solution drawings
59: 3 P-pentacubes, 3 L4-pentacubes
2 solutions for B, 2 solutions for H, 3 solutions for C, 5 solutions for K, 6 solutions for A, many solutions for D, many solutions for E, many solutions for F, many solutions for G and many solutions for J
3×
3×
solution drawings
60: 3 A-pentacubes, 3 L4-pentacubes
2 solutions for A, 2 solutions for F and 4 solutions for E
3×
3×
solution drawings
61: 3 P-pentacubes, 3 L2-pentacubes
2 solutions for A, 2 solutions for G, 4 solutions for F, 4 solutions for J, 5 solutions for D, 5 solutions for K and 8 solutions for E
3×
3×
solution drawings
62: 3 A-pentacubes, 3 L3-pentacubes
2 solutions for F and 3 solutions for A
3×
3×
solution drawings
63: 3 Y-pentacubes, 3 Q-pentacubes
2 solutions for G, 3 solutions for E, 8 solutions for B and many solutions for D
3×
3×
solution drawings
64: 3 P-pentacubes, 3 S-pentacubes
2 solutions for E, 4 solutions for G, 5 solutions for A and many solutions for D
3×
3×
solution drawings
65: 3 P-pentacubes, 3 N2-pentacubes
3 solutions for J
3×
3×
solution drawings
66: 3 L-pentacubes, 3 Q-pentacubes
3 solutions for G, 8 solutions for C, 8 solutions for F, many solutions for B, many solutions for D and many solutions for E
3×
3×
solution drawings
67: 3 L-pentacubes, 3 P-pentacubes
6 solutions for C, many solutions for A, many solutions for B, many solutions for D, many solutions for E, many solutions for F and many solutions for G
3×
3×
68: 3 U-pentacubes, 3 Q-pentacubes
8 solutions for A
3×
3×