The Mathematical Forest is grown in a two-dimensional plane, where trees can only grow on points with integer coordinates. To start with, there are no trees at all. The foresters plant the first tree at (0, 0). Each year, they carry out tree planting according to the following rule. If there is a tree on the point (m, n) but there are no trees on the points (m + 1, n) and (m, n + 1), then they can choose to remove the tree on (m, n) and plant new trees on the points (m, n + 1) and (m + 1, n). For an integer k ≥ 1, the kth diagonal consists of all points (m, n) with m + n = k − 1. Is it possible for the foresters to arrange their planting so that eventually there are no trees on the first 2 diagonals? What about the first 3 diagonals? 4 diagonals? Can you generalize?

Louise measured the perimeter of her rectangular scrapbook to be 154 cm. if the scrapbook is 45 cm wide, how long is the scrapbook?

Don't know how to find x given these segment lengths

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.