No, seriously, ARURF isn't random, or at least, doesn't make an attempt to feel random.

What I propose is this:

• All champions are unlocked
• The pool for both teams is drawn from the 33% least played champions.
• Rerolls would just dispense another champion from

This way, every champion will be seen, whilst still being fair for both teams. IN FACT, this could work for normal URF, just with a limited pool for both teams.

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Now, being fair? That's a whole other issue.