There are 100 instances located at points 1, 2, ..., 100 on the political spectrum (the real number line).
There are N users who move from instance to instance. There are no alt accounts: each user uses exactly one instance at each moment in time. Users may move from their current instance to any politically adjacent one. (This means that, starting at instance 5, they can move to 4 or 6.)
Eliza Fox wishes to destroy the fediverse, while Jeff Cliff wishes to protect it. At the start, Jeff chooses the starting distribution of users. Then there are 99 turns, each of which has three phases:
First, Eliza chooses an instance to Fediblock, destroying it completely.
Second, Jeff moves each user of the now-destroyed instance to any politically adjacent instance which has not been destroyed. (If no such instance exists, then those users are executed.)
Third, Jeff may move all users however much he wants (including the users that were already moved in the second phase), as long as they do not enter (or jump over) a destroyed instance.
At the end of the 99 turns, 99 instances have been destroyed, so there is only 1 instance remaining.
(For convenience, let us say that users can be split into fractions without harming them in any way.)
Your task: Explain how Jeff can save the lives of N/50 users. Furthermore, explain how Eliza can prevent him from saving more than N/50 lives.
Example: Suppose there are 4 instances and 40 users. First, Jeff will choose the starting distribution of users, and then Eliza and Jeff will take 3 turns.
Jeff chooses to distribute the users evenly, so the user count is (10, 10, 10, 10).
During the first turn, Eliza destroys instance 2, and Jeff moves 4 of those users to the left and 6 of them to the right. The user count is now (14, _, 16, 10). Jeff moves 3 users from instance 3 to instance 4. The user count is now (14, _, 13, 13). Note that Jeff cannot move any users from instance 1 to instance 3 or 4, because they would have to cross through instance 2, which no longer exists.
During the second turn, Eliza destroys instance 1. Since those 14 users have nowhere to go, they are executed. The user count is now (_, , 13, 13). Jeff moves 2 users from instance 4 to instance 3. The user count is now (, _, 15, 11).
During the third turn, Eliza destroys instance 3, and Jeff is forced to move all 15 of those users to the right. The user count is now (_, _, _, 26). Jeff has saved 26 out of 40 lives.
Extra credit: Let us make the model more realistic by requiring that, during the second phase of each turn, all users of the now-destroyed instance must move to the right. (Again, if this is not possible, then those users are executed.) Explain how Jeff can still save N/50 lives.
