Question 175
Let $n$ be a natural number. A staircase of size $n$ is made of small $1 \times 1$ squares, with 1 square for the first step, 2 squares for the second step, and so on, up to $n$ squares for the $n^{\text{th}}$ step.
We have square stones (with integer side) of all sizes to build this staircase, and we denote by $f(n)$ the minimum number of stones that must be used for a staircase of size $n$. For example, $f(2)=3$ and $f(4)=7$, as illustrated below.
1. Find all integers $n \geqslant 0$ such that $f(n)=n$.
2. Find all integers $n \geqslant 0$ such that $f(n)=n+1$
We have square stones (with integer side) of all sizes to build this staircase, and we denote by $f(n)$ the minimum number of stones that must be used for a staircase of size $n$. For example, $f(2)=3$ and $f(4)=7$, as illustrated below.
1. Find all integers $n \geqslant 0$ such that $f(n)=n$.
2. Find all integers $n \geqslant 0$ such that $f(n)=n+1$