Let c(1), c(2), . . . be a sequence of positive integers in which every positive integer appears exactly once. A one-way infinite sequence of cells is given, where for each positive integer n, the n-th cell from the left is labelled with c(n). At time t = 0, an invisible trolley (Figure 1) is placed on an unknown cell, namely the k0-th cell from the left. For each integer t ⩾ 1, the following events occurs in order:
(i) Bob chooses a direction, either left or right, and the invisible trolley moves one cell in the chosen direction, arriving at the kt-th cell from the left.
(ii) The invisible trolley sends Bob the word “Yes” if the label of the cell it currently occupies is strictly less than the current time (that is, if c(kt) < t), and the word “No” otherwise.
If at any point the invisible trolley falls off the sequence of cells (that is, if kt = 0), Bob immediately loses. At any time, Bob may guess the position of the invisible trolley. If the guess is correct, he safely retrieves the invisible trolley; if the guess is incorrect, he immediately loses. Determine all sequences (c(n))n⩾1 for which Bob has a strategy to ensure the safe retrieval of the invisible trolley.
Answer format: State for which sequences (c(n))n⩾1 Bob has a strategy to ensure the safe retrieval of the invisible trolley, and prove it.
Please note, we will only take your first submission into account.
