Tessellate S.T.E.M.S - Mathematics - School - Set 3 - Problem 4

Let $n \geq 3$ be a natural number.

$A := \big\{f :\{1,2,...,n\} \rightarrow \mathbb{N}\big\}$ , the set of all natural number valued functions on the set $\{1,2,...,n\}$ .

For an element of $A$ , we are allowed the following two operations:

• Pick $i,j \in \{1,2,...,n\}, \space i\neq j$ , and replace $f$ with $f_1$ such that $\displaystyle f_1(k) = \begin{cases} \frac{f(i)+f(j)}{2} \quad \text{if } k \in \{i,j\}\\ f(k) \quad \quad \space \text{otherwise}\end{cases}$

• Replace $f$ with $f_2$ such that $f_2(k) = f(k)+1 \space \space \forall \space k \in \{1,2,...,n\}$

We call a positive integer $m \space$ good if $\exists \space f \in A$ such that there is a sequence of operations which transforms $f$ into $F(i) = f(i) + mi \space \space \forall \space i \in \{1,2,...,n\}$ .

How many good integers are there?

---

This problem is a part of Tessellate S.T.E.M.S.