Consider n pennies arranged in a straight line.

A move consists of taking a penny and turning it over (from head to tail or visa versa) and of doing the same to each of its neighbors.

If the penny happens to be at the end of the line, then it will have only one neighbor. For example, if the sequence is HHTH and you choose the 3rd penny, then your move would result in HTHT.

The question is, given any sequence of n pennies, is it possible to find a sequence of moves to bring it to all Heads? Show that the answer is no for n=5 but yes for n=6.

Number the coins 1 to 6, in order.

Coin 1 can be flipped while leaving others unchanged in 4 moves, by flipping coins 2, 3, 5 and 6

Coin 2 can be flipped while leaving others unchanged in 5 moves, by flipping coins 1,2,3,5 and 6.

Coin 3 can be flipped while leaving others unchanged in 2 moves, by flipping coins 1 and 2.

Coin 4 can be flipped while leaving others unchanged in 2 moves, by flipping coins 5 and 6.

Coin 5 can be flipped while leaving others unchanged in 5 moves, by flipping coins 1,2,4,5 and 6.

Coin 6 can be flipped while leaving others unchanged in 4 moves, by flipping coins 1, 2, 4, and 5

Therefore, any arrangement can be reached from any other arrangement.