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-bot

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.

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