It is not possible to construct a magic square of cubes of order 6. Using modular arithmetic, we can show that it is impossible to obtain a sum of cubes that is a multiple of 6 or congruent to a multiple of 6 modulo 6.

If we cube both sides of the equation 6S ≡ 0 (mod 6), we get:

216S³ ≡ 0 (mod 6)

Simplifying, we get:

S³ ≡ 0 (mod 6)

Now, we can consider all possibilities for each value of the sum S:

• If S is a multiple of 6, then S³ is a multiple of 216, which means it is impossible to obtain a sum of cubes that is a multiple of 6.
• If S is a multiple of 3, but not of 6, then S³ is a multiple of 27, which means it is impossible to obtain a sum of cubes that is congruent to a multiple of 6 modulo 6.
• If S is not a multiple of 3, then S³ is congruent to 1 (mod 3) or -1 (mod 3). This means we cannot obtain a sum of cubes that is congruent to a multiple of 6 modulo 6.

Therefore, it is not possible to construct a magic square of cubes of order 6.