Consider a game where you have five distinct coins in a row, and you are allowed...

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Consider a game where you have five distinct coins in a row, and you are allowed to move them in pairs and insert them between other coins in the row (without changing the order of the pair). For example, here is an allowable move: D As above you can slide the coins over horizontally to create space to insert the pair, or to remove extra space, as long as you do not change the order of the coins in the row. (a) Suppose the coins are placed in order of increasing value; is there a sequence of moves as described above that reverses the order (so they are in order of decreasing value)? (b) Again, suppose the coins are placed in order of increasing value; is there a sequence of moves that switches the first coin with the last, and where the middle three end up in their starting positions? Consider a game where you have five distinct coins in a row, and you are allowed to move them in pairs and insert them between other coins in the row (without changing the order of the pair). For example, here is an allowable move: D As above you can slide the coins over horizontally to create space to insert the pair, or to remove extra space, as long as you do not change the order of the coins in the row. (a) Suppose the coins are placed in order of increasing value; is there a sequence of moves as described above that reverses the order (so they are in order of decreasing value)? (b) Again, suppose the coins are placed in order of increasing value; is there a sequence of moves that switches the first coin with the last, and where the middle three end up in their starting positions?

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