AMC Practice #1 - Problem 15

Munch has 20 gold coins and 20 silver coins placed in a row in some random order. Define a block as a subsequence of any adjacent coins which are the same. For a value of k (1 \le k \le 40), Munch performs the following operation: He finds the biggest block which contains the kth coin from the left, and moves that block to the start. For example, if the row was \text{GGGSSGSS}, and k = 5, then the new row becomes \text{SSGGGGSS}.

Jerry Yang saw Munch’s brilliant idea, and got jealous. Jerry now has has n gold coins and n silver coins placed in a row in some random order. For a value of l (1 \le l \le 2n), Jerry performs the following operation: He finds the biggest block which contains the lth coin from the left, and takes it out. He then reverses the order of the row, and adds the block back at the start. For example, if the row was \text{GGGSSGSS}, and l = 5, then the new row becomes \text{SSSSGGGG}.

Let M be the sum of all values of k such that for all initial arrangement of the coins, after some number of operations, Munch will have all the gold coins in 1 block, and all the silver coins in another block. Let J be the sum of all values of l such that for all initial arrangement of the coins, after some number of operations, Jerry will have all the gold coins in 1 block, and all the silver coins in another block. Find the minimum value of n such that J \ge M.