Suppose we have two coins, one fair and the other with probability p of heads and 1-p of tails. Let X0, X1, X2, .... be the random walk constructed as follows. Put X0 = 0, and for n > 0, toss one of the coins to decide whether Xn is one greater (heads) or one less than Xn - 1. For the first jump (n= 1),make this toss with the fair coin. For the other jumps, toss the fair coin if min{Xk : k < n} < Xn-1 < max{Xk : k < n}, and otherwise toss the p coin. In other words, only use the p coin if the walker is at the edge of the interval of visited sites. If p is not equal to 1/2, the jump probabilities depend on the history of the walk, so the walk is not Markov, and many of the tools used to study fair random walk (the p = 1/2 case) are not applicable. I will discuss what is known about these and some related random walks.