Poker Chip Swap
My Dad tought me a game back in the 60's (which Mark Thompson tells me was also published by Martin Gardner in one of his essays) that also was the inspiration for Mark's "Three Penny Operations" puzzle published in Games magazine (Feb 99, page 64). The way my Dad showed it to me (which was the way that HIS Dad had shown it to him) was:
Take N>=3 RED and WHITE poker chips, and line them up with the WHITE chips all on the left and the RED chips all on the right, thusly:
Then, moving two adjacent chips at a time, in N moves convert the starting position into alternating RED and WHITE:
"Two adjacent chips" means two chips that are side by side, with no gaps between them. So, the first move has to be to move one of the five possible pairs of adjacent chips to either the left end or right end of the line of chips. As you proceed, you will frequently have a "two chip gap" in the middle of the line of chips, and at that point you can move a pair into that gap as well as to either end of the line.
You can use anything for markers, if you don't have poker chips. Pennies and nickels or dimes work well and are readily available.
I have solutions for N from 3 through 10. 3 is kind of a special case, but from there on up, things start to fall into a pattern of sorts, so I suspect that ALL cases >=3 are solvable in this way. For solutions, click here.