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Solved by 127 teams.

First solved after 27 minutes.

Shortest team solution: 876 bytes.

Shortest judge solution: 793 bytes.

Start by considering only one gene type i. If the number of si and ei differs, then gene type i clearly cannot form properly a nested sequences, so we will ignore those and only focus on the i with the same number of si and ei .

Now consider reading clockwise from the cut point, incrementing a counter at every si and decrementing it at every ei . The sequence will be properly nested if and only if the lowest value of the counter occurs at the cut point (possibly tied). We can start with an arbitrary cut point and a counter for each gene type initialized to zero, and determine the minimum counter value for each gene type. This will tell us the score for that initial cut point. We can now incrementally move the cut point around the circle: each time it moves, one gene type might become properly nested or cease to be properly nested, depending on the new counter value for the type of the marker we just moved past.