In a 4-column × 6-row grid (24 slots), an inversion occurs whenever a larger tile number precedes a smaller tile number in reading order. Because the grid width (4) is even, moving the blank up or down a row changes the inversion count by an odd number (±1 or ±3). A state is solvable if and only if (inversions + rowOfBlankFromBottom) has the same even/odd parity as the solved state!

Most players rely on habitual horizontal/vertical scanning. Tilting the 24-slot field onto its diagonal axis forces the brain's parietal cortex to perform active mental rotation—the exact skill that predicts future STEM invention.