Question 11
As shown in the diagram below, a frog jumps along a triangular spiral path, starting at point $P_0$ (which represents the number $0$). The path consists of straight segments whose lengths increase by one unit after each turn: the first segment has length $1$, the second has length $2$, the third has length $3$, and so on.
Each jump moves the frog by exactly one unit along the path to the next numbered point. (For example, jump $1$ lands on $1$, jump $2$ lands on $2$, jump $3$ lands on $3$, and so on.) Following this pattern, the frog reaches point $P_1$ (position $6$) after $6$ jumps and point $P_2$ (position $21$) after $21$ jumps.
Let $P_n$ denote the $n$th point along the vertical line shown. How many jumps must the frog make to travel starting from $P_0$ to point $P_{14}$?
Each jump moves the frog by exactly one unit along the path to the next numbered point. (For example, jump $1$ lands on $1$, jump $2$ lands on $2$, jump $3$ lands on $3$, and so on.) Following this pattern, the frog reaches point $P_1$ (position $6$) after $6$ jumps and point $P_2$ (position $21$) after $21$ jumps.
Let $P_n$ denote the $n$th point along the vertical line shown. How many jumps must the frog make to travel starting from $P_0$ to point $P_{14}$?