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    $\begingroup$ In cases 2&3 you completely rearrange some people from the existing groups, forming completely new groups, but don’t specify how to place all the people from the old groups into the new groups, let alone prove this is possible. You’re basically saying, start from scratch putting yourself in one group first and your neighbours in the other groups Why would that work out with the rest of the people from the old groups and not need you to rearrange everything again? $\endgroup$ Commented yesterday