Clingo, 220 + 2 = 222 bytes
1{s(S):S=0..n,S**3==n}.
A{a(0..n-1,A)}:-A=1..n.
:-a(P,A-1),{a(P,A)}0,A<n.
:-s(S),{a(P,A):P-P/K\S*K=Q-Q/K\S*K}>(S*n+S)/2,Q=0..n,K=(1;S;S*S).
:-s(S),a(((0;1)+(0;S)+(0;S*S))*~-S,A),{a(0,A)}0.
:-s(S),a((S;S*S)*~-S,A),{a(S-1,A)}0.
Run with clingo magic-cubes.lp -n0 -cn=INPUT
. The answer will be printed next to Models
.
All newlines are optional and shown here for clarity but excluded from the byte count. I’m scoring +2 bytes for -n0
(“count all models”).
The last two lines ensure that only one rotated version of each cube is counted.
The answers for 1, 8, 27 are quickly found to be 1, 0, 216; the rest are probably too hard.
If you want to see the cubes in a more readable format, add b((P\S,P/S\S,P/S/S),A):s(S),-a(P,A),not a(P,A-1). #show b/2.
$ clingo magic-cubes.lp -n0 -cn=1
clingo version 5.1.0
Reading from magic-cubes.lp
Solving...
Answer: 1
a(0,1) s(1)
SATISFIABLE
Models : 1
Calls : 1
Time : 0.009s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time : 0.010s
$ clingo magic-cubes.lp -n0 -cn=8
clingo version 5.1.0
Reading from magic-cubes.lp
Solving...
UNSATISFIABLE
Models : 0
Calls : 1
Time : 0.010s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time : 0.000s
$ clingo magic-cubes.lp -n0 -cn=27
clingo version 5.1.0
Reading from magic-cubes.lp
Solving...
Answer: 1
a(7,1) a(7,2) a(13,2) a(7,3) a(13,3) a(19,3) a(7,4) a(13,4) a(15,4) a(19,4) a(7,5) a(13,5) a(15,5) a(19,5) a(21,5) a(0,6) a(7,6) a(13,6) a(15,6) a(19,6) a(21,6) a(0,7) a(7,7) a(13,7) a(15,7) a(19,7) a(21,7) a(26,7) a(0,8) a(5,8) a(7,8) a(13,8) a(15,8) a(19,8) a(21,8) a(26,8) a(0,9) a(5,9) a(7,9) a(11,9) a(13,9) a(15,9) a(19,9) a(21,9) a(26,9) a(0,10) a(2,10) a(5,10) a(7,10) a(11,10) a(13,10) a(15,10) a(19,10) a(21,10) a(26,10) a(0,11) a(2,11) a(5,11) a(7,11) a(11,11) a(13,11) a(15,11) a(17,11) a(19,11) a(21,11) a(26,11) a(0,12) a(2,12) a(5,12) a(7,12) a(11,12) a(13,12) a(15,12) a(17,12) a(19,12) a(21,12) a(23,12) a(26,12) a(0,13) a(2,13) a(5,13) a(7,13) a(10,13) a(11,13) a(13,13) a(15,13) a(17,13) a(19,13) a(21,13) a(23,13) a(26,13) a(0,14) a(2,14) a(5,14) a(7,14) a(10,14) a(11,14) a(13,14) a(15,14) a(17,14) a(19,14) a(21,14) a(23,14) a(25,14) a(26,14) a(0,15) a(2,15) a(4,15) a(5,15) a(7,15) a(10,15) a(11,15) a(13,15) a(15,15) a(17,15) a(19,15) a(21,15) a(23,15) a(25,15) a(26,15) a(0,16) a(2,16) a(4,16) a(5,16) a(7,16) a(10,16) a(11,16) a(13,16) a(15,16) a(17,16) a(18,16) a(19,16) a(21,16) a(23,16) a(25,16) a(26,16) a(0,17) a(2,17) a(4,17) a(5,17) a(6,17) a(7,17) a(10,17) a(11,17) a(13,17) a(15,17) a(17,17) a(18,17) a(19,17) a(21,17) a(23,17) a(25,17) a(26,17) a(0,18) a(2,18) a(4,18) a(5,18) a(6,18) a(7,18) a(10,18) a(11,18) a(12,18) a(13,18) a(15,18) a(17,18) a(18,18) a(19,18) a(21,18) a(23,18) a(25,18) a(26,18) a(0,19) a(2,19) a(3,19) a(4,19) a(5,19) a(6,19) a(7,19) a(10,19) a(11,19) a(12,19) a(13,19) a(15,19) a(17,19) a(18,19) a(19,19) a(21,19) a(23,19) a(25,19) a(26,19) a(0,20) a(2,20) a(3,20) a(4,20) a(5,20) a(6,20) a(7,20) a(9,20) a(10,20) a(11,20) a(12,20) a(13,20) a(15,20) a(17,20) a(18,20) a(19,20) a(21,20) a(23,20) a(25,20) a(26,20) a(0,21) a(2,21) a(3,21) a(4,21) a(5,21) a(6,21) a(7,21) a(9,21) a(10,21) a(11,21) a(12,21) a(13,21) a(15,21) a(17,21) a(18,21) a(19,21) a(21,21) a(23,21) a(24,21) a(25,21) a(26,21) a(0,22) a(2,22) a(3,22) a(4,22) a(5,22) a(6,22) a(7,22) a(9,22) a(10,22) a(11,22) a(12,22) a(13,22) a(14,22) a(15,22) a(17,22) a(18,22) a(19,22) a(21,22) a(23,22) a(24,22) a(25,22) a(26,22) a(0,23) a(2,23) a(3,23) a(4,23) a(5,23) a(6,23) a(7,23) a(9,23) a(10,23) a(11,23) a(12,23) a(13,23) a(14,23) a(15,23) a(17,23) a(18,23) a(19,23) a(20,23) a(21,23) a(23,23) a(24,23) a(25,23) a(26,23) a(0,24) a(2,24) a(3,24) a(4,24) a(5,24) a(6,24) a(7,24) a(8,24) a(9,24) a(10,24) a(11,24) a(12,24) a(13,24) a(14,24) a(15,24) a(17,24) a(18,24) a(19,24) a(20,24) a(21,24) a(23,24) a(24,24) a(25,24) a(26,24) a(0,25) a(2,25) a(3,25) a(4,25) a(5,25) a(6,25) a(7,25) a(8,25) a(9,25) a(10,25) a(11,25) a(12,25) a(13,25) a(14,25) a(15,25) a(17,25) a(18,25) a(19,25) a(20,25) a(21,25) a(22,25) a(23,25) a(24,25) a(25,25) a(26,25) a(0,26) a(1,26) a(2,26) a(3,26) a(4,26) a(5,26) a(6,26) a(7,26) a(8,26) a(9,26) a(10,26) a(11,26) a(12,26) a(13,26) a(14,26) a(15,26) a(17,26) a(18,26) a(19,26) a(20,26) a(21,26) a(22,26) a(23,26) a(24,26) a(25,26) a(26,26) a(0,27) a(1,27) a(2,27) s(3) a(3,27) a(4,27) a(5,27) a(6,27) a(7,27) a(8,27) a(9,27) a(10,27) a(11,27) a(12,27) a(13,27) a(14,27) a(15,27) a(16,27) a(17,27) a(18,27) a(19,27) a(20,27) a(21,27) a(22,27) a(23,27) a(24,27) a(25,27) a(26,27)
Answer: 2
[…]
Answer: 216
a(0,2) a(12,1) a(12,2) a(0,3) a(12,3) a(24,3) a(0,4) a(12,4) a(22,4) a(24,4) a(0,5) a(10,5) a(12,5) a(22,5) a(24,5) a(0,6) a(7,6) a(10,6) a(12,6) a(22,6) a(24,6) a(0,7) a(5,7) a(7,7) a(10,7) a(12,7) a(22,7) a(24,7) a(0,8) a(5,8) a(7,8) a(10,8) a(12,8) a(20,8) a(22,8) a(24,8) a(0,9) a(5,9) a(7,9) a(10,9) a(12,9) a(17,9) a(20,9) a(22,9) a(24,9) a(0,10) a(5,10) a(7,10) a(10,10) a(12,10) a(16,10) a(17,10) a(20,10) a(22,10) a(24,10) a(0,11) a(4,11) a(5,11) a(7,11) a(10,11) a(12,11) a(16,11) a(17,11) a(20,11) a(22,11) a(24,11) a(0,12) a(4,12) a(5,12) a(7,12) a(10,12) a(12,12) a(16,12) a(17,12) a(19,12) a(20,12) a(22,12) a(24,12) a(0,13) a(4,13) a(5,13) a(7,13) a(10,13) a(12,13) a(16,13) a(17,13) a(19,13) a(20,13) a(22,13) a(24,13) a(26,13) a(0,14) a(4,14) a(5,14) a(7,14) a(10,14) a(12,14) a(14,14) a(16,14) a(17,14) a(19,14) a(20,14) a(22,14) a(24,14) a(26,14) a(0,15) a(2,15) a(4,15) a(5,15) a(7,15) a(10,15) a(12,15) a(14,15) a(16,15) a(17,15) a(19,15) a(20,15) a(22,15) a(24,15) a(26,15) a(0,16) a(2,16) a(4,16) a(5,16) a(6,16) a(7,16) a(10,16) a(12,16) a(14,16) a(16,16) a(17,16) a(19,16) a(20,16) a(22,16) a(24,16) a(26,16) a(0,17) a(2,17) a(4,17) a(5,17) a(6,17) a(7,17) a(10,17) a(12,17) a(14,17) a(16,17) a(17,17) a(19,17) a(20,17) a(21,17) a(22,17) a(24,17) a(26,17) a(0,18) a(2,18) a(4,18) a(5,18) a(6,18) a(7,18) a(9,18) a(10,18) a(12,18) a(14,18) a(16,18) a(17,18) a(19,18) a(20,18) a(21,18) a(22,18) a(24,18) a(26,18) a(0,19) a(2,19) a(4,19) a(5,19) a(6,19) a(7,19) a(9,19) a(10,19) a(11,19) a(12,19) a(14,19) a(16,19) a(17,19) a(19,19) a(20,19) a(21,19) a(22,19) a(24,19) a(26,19) a(0,20) a(2,20) a(4,20) a(5,20) a(6,20) a(7,20) a(8,20) a(9,20) a(10,20) a(11,20) a(12,20) a(14,20) a(16,20) a(17,20) a(19,20) a(20,20) a(21,20) a(22,20) a(24,20) a(26,20) a(0,21) a(2,21) a(4,21) a(5,21) a(6,21) a(7,21) a(8,21) a(9,21) a(10,21) a(11,21) a(12,21) a(14,21) a(16,21) a(17,21) a(19,21) a(20,21) a(21,21) a(22,21) a(23,21) a(24,21) a(26,21) a(0,22) a(2,22) a(4,22) a(5,22) a(6,22) a(7,22) a(8,22) a(9,22) a(10,22) a(11,22) a(12,22) a(14,22) a(16,22) a(17,22) a(18,22) a(19,22) a(20,22) a(21,22) a(22,22) a(23,22) a(24,22) a(26,22) a(0,23) a(2,23) a(4,23) a(5,23) a(6,23) a(7,23) a(8,23) a(9,23) a(10,23) a(11,23) a(12,23) a(14,23) a(15,23) a(16,23) a(17,23) a(18,23) a(19,23) a(20,23) a(21,23) a(22,23) a(23,23) a(24,23) a(26,23) a(0,24) a(2,24) a(3,24) a(4,24) a(5,24) a(6,24) a(7,24) a(8,24) a(9,24) a(10,24) a(11,24) a(12,24) a(14,24) a(15,24) a(16,24) a(17,24) a(18,24) a(19,24) a(20,24) a(21,24) a(22,24) a(23,24) a(24,24) a(26,24) a(0,25) a(1,25) a(2,25) a(3,25) a(4,25) a(5,25) a(6,25) a(7,25) a(8,25) a(9,25) a(10,25) a(11,25) a(12,25) a(14,25) a(15,25) a(16,25) a(17,25) a(18,25) a(19,25) a(20,25) a(21,25) a(22,25) a(23,25) a(24,25) a(26,25) a(0,26) a(1,26) a(2,26) a(3,26) a(4,26) a(5,26) a(6,26) a(7,26) a(8,26) a(9,26) a(10,26) a(11,26) a(12,26) a(14,26) a(15,26) a(16,26) a(17,26) a(18,26) a(19,26) a(20,26) a(21,26) a(22,26) a(23,26) a(24,26) a(25,26) a(26,26) a(0,27) a(1,27) a(2,27) s(3) a(3,27) a(4,27) a(5,27) a(6,27) a(7,27) a(8,27) a(9,27) a(10,27) a(11,27) a(12,27) a(13,27) a(14,27) a(15,27) a(16,27) a(17,27) a(18,27) a(19,27) a(20,27) a(21,27) a(22,27) a(23,27) a(24,27) a(25,27) a(26,27)
SATISFIABLE
Models : 216
Calls : 1
Time : 6.101s (Solving: 6.07s 1st Model: 1.03s Unsat: 0.37s)
CPU Time : 6.090s
9+31+60=100≠106=34+43+29
. \$\endgroup\$ – PurkkaKoodari Jun 14 '17 at 8:13