Question number 29. A gardener wants to plant 100 trees, oaks and birches along an avenue in the park. The number of trees between any two oaks must not be equal to five. Of these 100 trees, what is the greatest number of oaks that the gardener can plant? Let's begin by making an observation about the number of oaks we can plant in a row. We know that any two oaks need to be separated by six birches. So if I have six birches in a row, then I can start planting oaks on either side, like so. And the gap between the oaks is at least six, so that's good. But if I plant more than six oaks in a row, then the gap between the oaks will be at least five. So far, with six in a row, it's only four. So that is one way to do the planting. But then we observe that in any row of 12, there are at most six oaks. For the same reason, if I have any two of them, I must separate them by six birches, and then the remaining trees can be oaks. So we have in a row of 100 trees, exactly eight groups of 12, and the last four can be oaks. So there are six times eight plus four, or 52 oaks that can be planted out of the 100 trees.

gardener

trees

oaks

birches

planting