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The following Python program does a breadth first search to find the minimum number of flips required to reach all possible outcomes from the starting position. It runs in 4.3s (2.0s under PyPy).

Solution:The minimum number of counters turned over is 33.You can also use this program to find the minimum number of flips required to reach other representations of dice faces (1 → 0 flips, 2 → 11 flips, 3 → 22 flips, 4 → 33 flips, 5 → 12 flips, 6 → 13 flips).