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Question Number 227529 by MrAjder last updated on 08/Feb/26
$$ \\ $$There are cards in 30 different colors, with 70 cards of each color. Initially, 70 cards are selected and laid out in a stack from top to bottom. One operation consists of: choosing an X from the bottom 20 cards, such that its color has not appeared in the top 50 cards; then choosing a Y from the top 50 cards, such that its color has appeared at least twice in the top 50 cards; removing X and placing it right above Y. Repeat this process until no more valid X can be chosen. (1) Prove that the number of operations is finite. (2) For all possible initial configurations and sequences of operations, find the maximum possible number of operations.