Aside from reducing the sets to be solved by excluding symmetries they had to brute force it. The math is rock hard.
The first two papers were simply worked out by hand. Make a somewhat clean technique for solving the cube, plug in a worst case number of turns for each step, show the cumulative worst case number. If it is lower than anything before it that’s the new upper bound for god’s number.
To keep pushing down the upper bounds was a problem that required brute force and they were throwing that at it back in 1992.
