Problem of the Month (June 2014)

This month's problem comes from George Sicherman.

A polycube is a solid made from gluing unit cubes together along their faces. If S is a set of positive integers, we call a polycube P a S-faced polycube if S is the set of areas of the faces of P. What is the smallest (in terms of volume) S-faced polycube for various S?

When S={n}, solutions are known for for all n except n = 2 and n = 3. Can you find a solution for one of these values of n? Can you prove they don't exist? Can you find smaller solutions for the other values of n?

Solutions are known for all S={m,n}. Can you improve any of the best known solutions? Is every S={m,n,p} possible?

Given two particular polyominoes, is there a polycube that only has those two faces?

If there are an equal number n of faces with each area in S, we call P a balanced S-faced polycube. What balanced polycubes exist? In particular, what is the smallest k or the smallest n for which a {1,2,3,...k}n balanced polycube exists?

ANSWERS

Answers were received from George Sicherman, Alexey Nigin, Joe DeVincentis, and Bryce Herdt.

Smallest Known Solutions for S={n}
123456

1 		none
(Joe DeVincentis) 		?
8
25
(George Sicherman)
288
(George Sicherman)

7891011

245
(Bryce Herdt)
64
27
500
(Bryce Herdt)
121

121314151617181920

144
169
196
225
64
289
324
361
200

Smallest Known Solutions for S={m,n}
m \ n123456789
2
2
3
3
13
(George Sicherman)
4
4
4
15
(George Sicherman)
5
5
10
12
20
6
6
12
9
12
30
7
7
28
(George Sicherman)
24
(George Sicherman)
36
(Bryce Herdt)
18
72
(George Sicherman)
8
8
16
21
(George Sicherman)
16
40
48
48
(George Sicherman)
9
9
18
9
32
(Bryce Herdt)
45
18
24
53
(George Sicherman)
10
10
20
33
(George Sicherman)
20
25
(George Sicherman)
40
(Bryce Herdt)
91
80
90

We can also ask for solutions that have the same number of faces with area m and n.

Smallest Known Balanced Solutions for S={m,n}
m \ n12345
2
10
(George Sicherman)
3
6
(George Sicherman)
18
(George Sicherman)
4
30
(Bryce Herdt)
24
(Bryce Herdt)
15
(George Sicherman)
5
28
(Bryce Herdt)
164
(George Sicherman)
12
108
(George Sicherman)
6
42
(Bryce Herdt)
12
(George Sicherman)
18
(George Sicherman)
24
365
(George Sicherman)
7
48
(Bryce Herdt)
469
(George Sicherman)
126
(Bryce Herdt)
253
(George Sicherman)
18
8
25
(Bryce Herdt)
208
(Bryce Herdt)
115
(Bryce Herdt)
32
(Bryce Herdt)
323
(George Sicherman)
9
26
(Bryce Herdt)
379
(George Sicherman)
90
(Bryce Herdt)
131
(George Sicherman)
46
10
62
(George Sicherman)
276
(George Sicherman)
280
(Bryce Herdt)
156
(George Sicherman)
50
(George Sicherman)

m \ n6789
7
319
(George Sicherman)
8
436
(George Sicherman)
412
(Bryce Herdt)
9
198
(Bryce Herdt)
24
53
(George Sicherman)
10
224
(George Sicherman)
476
(George Sicherman)
584
(Bryce Herdt)
684
(Bryce Herdt)

Smallest Known Solutions for S={1,m,n}
m \ n23456789
3
3
(GS)
4
4
(GS)
4
(GS)
5
5
(GS)
6
(GS)
8
(GS)
6
6
(GS)
6
(GS)
7
(GS)
10
(GS)
7
7
(GS)
7
(GS)
8
(GS)
7
(GS)
11
(GS)
8
8
8
8
9
(GS)
9
11
(GS)
9
9
9
9
9
10
(GS)
10
13
(GS)
10
10
(GS)
10
(GS)
10
(GS)
10
(GS)
10
(GS)
10
(GS)
11
(GS)
14
(GS)

Smallest Known Solutions for S={2,m,n}
m \ n3456789
4
6
(GS)
5
8
(GS)
10
(GS)
6
6
(GS)
8
(GS)
18
(GS)
7
13
(GS)
14
(GS)
12
(GS)
14
(GS)
8
16
(GS)
8
29
(BH)
16
28
(GS)
9
14
(GS)
18
(BH)
19
(BH)
18
(GS)
16
(GS)
18
(BH)
10
16
(GS)
16
(GS)
10
(GS)
12
(GS)
14
(GS)
20
(GS)
18
(GS)

Smallest Known Solutions for S={3,m,n}
m \ n456789
5
17
(BH)
6
12
(GS)
15
(GS)
7
19
(BH)
14
(GS)
21
8
12
(GS)
20
(GS)
24
42
(GS)
9
12
(GS)
18
(BH)
18
18
(BH)
18
(BH)
10
22
(BH)
15
(BH)
30
(GS)
30
(BH)
27
(BH)
30
(GS)

Smallest Known Solutions for S={4,m,n}
m \ n56789
6
24
(GS)
7
25
(BH)
42
(GS)
8
20
24
28
9
19
(GS)
30
(GS)
44
(BH)
32
(GS)
10
20
(BH)
24
(GS)
40
(BH)
32
(GS)
40
(BH)

Smallest Known Solutions for S={5,m,n}
m \ n6789
7
34
(BH)
8
58
(GS)
60
(BH)
9
36
(GS)
20
(BH)
59
(BH)
10
30
(BH)
35
(BH)
40
(BH)
45
(BH)

Smallest Known Solutions for S={6,m,n}
m \ n789
8
40
(BH)
9
36
(GS)
36
(GS)
10
54
(BH)
32
(BH)
60
(GS)

Smallest Known Solutions for S={7,m,n} and {8,m,n}
{7,8,9}{7,8,10}{7,9,10}{8,9,10}

46
(BH)
112
(GS)
119
(BH)
52
(BH)

Bryce Herdt managed to prove that all triples {m,n,p} have a solution. Then, in 2021, Bryce Herdt and George Sicherman proved that any set of two or more positive integers is the set of different areas of the faces of some polycube!

Smallest Known Solutions for Two Given Faces

2
(GS)

3
(GS) 		?

4
(GS)
13
(GS) 		?

4
(GS) 		? ? ?
?
16 		? ? ?

48
(GS)
4
(GS) 		?
15
(GS) 		?
24
(GS)

5
(GS)
20
(GS)
15
(GS) 		? ? ? ?

8
(GS) 		? ? ? ? ? ? ?

5
(GS) 		? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?

30
(GS) 		?
18
(GS) 		? ? ? ? ? ?

14
(GS) 		? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?

38
(BH) 		? ? ? ? ? ? ? ?

14
(GS) 		? ?
12 		? ?
30
(GS) 		? ?

9
(GS)
36
(GS) 		? ? ? ? ? ? ?

5
(GS)
10
15 		?
20 		? ? ? ?

10
(GS) 		? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?

Smallest Known k with Faces with Area 1-k,
Each Occurring n Times
n=1
k=11

19
(AN) 		n=2
k=5

7
(GS) 		n=3
k=4

7
(GS) 		n=4
k=4

10
(GS) 		n=5
k=4

12
(AN) 		n=6
k=1

1 		n=7
k=? 		n=8
k=3

12
(GS)

Smallest Known n with Faces with Area 1-k,
Each Occurring n Times
k=1
n=6

1 		k=2
n=14

10
(GS) 		k=3
n=6

9
(GS) 		k=4
n=3

7
(GS) 		k=5
n=2

7
(GS) 		k=6
n=2

10
(GS)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/2/14.