How many Magic Cubes?

Question

Your Task:

You all know what a magic square is! But what about a magic cube? A magic cube is the 3D version of a magic square, and is a set of numbers in a cube, where the sums of every straight (not diagonal) line are equal. You must write a program or function that receives an integer, and returns the number of valid magic cubes made up of up to one of each integer between 1 and that integer (inclusive).

Input:

An integer.

Output:

The number of magic cubes you can make by using up to one of of each integer between 1 and the input. A cube with various rotations should only be counted once.

Scoring:

This is code-golf, lowest byte count wins.

BOUNTY:

I will award a 100 rep bounty to the first user that manages to do this in < 100 bytes without using built-ins designed specifically towards magic cubes or squares (looking at you Mathematica).

Answer 1

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

Answer 2

Ruby, 275 bytes

n=gets.to_i;p (2.upto(n**(1/3r)).map{|i|(1..n).to_a.permutation(i**3).map{|x|c=eval'x'+".each_slice(#{i}).to_a"*2;[c,c.map(&:transpose),c.transpose.map(&:transpose)].map{|q|q.map{|a|a.map{|b|b.inject(:+)}}.flatten.uniq.length}.inject(:+)==3?1:0}}+[0]).flatten.inject(:+)/48+n

Try it online!

In theory, this works. In practice, since it creates every permutation of the list of possible numbers, it'll take hopelessly large amounts of time and memory for any input more than 13 or so. I haven't taken the time to test it past 12, but it seems no 2×2×2 magic cubes exist for that range, so all it really does is tell you how many 1×1×1 cubes you can make.

I "golfed" it a bit (really just made it hard to read for minimal size reduction returns). I'm sure the size can be reduced quite a bit, though getting it under 100 characters is a pretty tall order.

Ruby 2.4 added .sum, but TIO doesn't support it yet, so this has a few .inject(:+)s.

I'll add a non-golfed version + explanation if requested.

Answer 3

Jelly, 36 bytes

;Z;Z€ẎS€E
*3³œcŒ!€Ẏs€s€¹Ç€S
Ç€ḊS÷24+

Try it online!

As you expect, it's terribly slow.

Explanation

• Link 1: Given a cube, check if it's valid.

;Z;Z€ẎS€E   Input: N =
            [  [ [a,b],    [ [e,f],
                 [c,d] ],    [g,h] ]   ]

;Z          Concatenate with zip of N. The zip (transpose) of N is
            [   [ [a,b],    [ [c,d],
                  [e,f] ],    [g,h] ]   ]
            (by treating each 2-element block as an unit
            and transpose the large table)

;Z€         Concatenate with plane-wise zip of N, that is
            [  [ [a,c],    [ [e,g],
                 [b,d] ],    [f,h] ]   ]
            (transpose each 2x2 block)

            The current value is:
            [ [ [a,b],   [ [e,f],   [ [a,b],   [ [c,d],   [ [a,c],   [ [e,g],
                 [c,d] ],   [g,h] ],   [e,f] ],   [g,h] ],   [b,d] ],   [f,h] ] ]

Ẏ           Tighten, concatenate all elements inside, similar to `;/`.
            Get [ [a,b], [c,d], [e,f], [g,h],
                  [a,b], [e,f], [c,d], [g,h],
                  [a,c], [b,d], [e,g], [f,h] ]
S€          Sum each sublist. `ZS` also works.
E           All Equal? `1` if all elements are equal, `0` otherwise.

• Link 2: Given a side length x, generate all cubes with that side length and calculate number of valid cubes.

*3³œcŒ!€Ẏs€s€¹Ç€S   Given: `x` (side length).
*3                  Calculate x³.
  ³                 First commandline argument. (n)
   œc               Generate all subsets of the range `[1..n]` having length x³.
     Œ!€Ẏ           All permutation of each and tighten.
                      (the current value is a list of all ways to choose
                      an ordered tuple of x³ elements from `[1..n]`)
         s€s€       Split each twice into slices of length `x`.
                      On a list of (list of x³ integers), this format
                      each sublist into a cube.
             ¹      Do nothing. Because Jelly is a tacit programming language
                      without this the program will behave differently.
              Ç€    Apply last link for €ach.
                S   Sum.

• Link 3 (main link): For all side lengths from 1 to n, calculate number of cubes (rotations are counted multiple times) for each side length, dequeue and sum it (so we get the sum of number of cubes with side length >= 2), divide by 24 (to avoid duplication of cubes) and add the number of side length 1 cube (input).

Ç          Apply last link...
 €           for €ach element from `1` to `n`.
  Ḋ        Ḋequeue, dispose the result for value 1.
   S÷24    Sum and divide by 24.
       +   Add the input (number of cubes with side length 1)