Problem 11 · Trees and agaves
Each phase doubles the gaps: $4 \to 8 \to 16$ gaps.
Integer answer, at most 4 digitsAlong a straight path, $5$ trees are initially planted, far apart from each other. In a second phase, another tree is planted midway between each pair of adjacent trees. They still seem too far apart, so again a tree is planted midway between every two adjacent trees. Finally, an agave is planted between every two adjacent trees. How many agaves have been planted?
Copa Cangur · SCM
Easy
Closed answer
Reasoned solution
Key idea: planting midway between each consecutive pair adds as many trees as there are gaps, and the number of gaps doubles.
With $5$ trees there are $4$ gaps. Second phase: $+4$ trees → $9$ trees and $8$ gaps. Third phase: $+8$ trees → $17$ trees and $16$ gaps.
$$\text{Agaves} = \text{final gaps} = 16.$$
Answer: 16