Surely given enough time, you'd solve it by accident? A bit like the explanation of infinity, where if you had infinitely many monkeys with type writers, one of them would type a line of Shakespeare at some point.
For a 3x3x3 cube, there are 43,252,003,274,489,856,000 permutations. So, probably not!
The 7x7x7 cube has (ready for this..?) 19,500,551,183,731,307,835,329,126,754,019,748,794,904,992,692,043,434,567,152,132,912,323,232,706,135,469,180,065,278,712,755,853,360,682,328,551,719,137,311,299,993,600,000,000,000,000,000,000,000,000,000,000,000 permutations

