Problem of the Month
Hint for Problem 6: March 2023

1. There is no hint given for this part, but it might be useful in later parts to see if you notice any patterns in the distribution of the possible outputs of the function $f$.

2. See part (a).

3. If $n^2<m<(n+1{)}^2$, then $f(m)=d$ is equivalent to $n+{\displaystyle \frac{d}{10}}<\sqrt{m}<n+{\displaystyle \frac{d+1}{10}}$.

4. Similar to the result in (c), if $n$ is one more than a multiple of $5$, then in the list $$f(n^2+1),f(n^2+2),\dots ,f(n^2+2n)$$ every possible value from $0$ through $9$ appears exactly ${\displaystyle \frac{n-1}{5}}$ times, with the exception of $4$ and $7$ which appear ${\displaystyle \frac{n-1}{5}}+1$ times each. Try to find and prove other similar results.