Question 137
Call a year ultra-even if all of its digits are even. Thus $2000$, $2002$, $2004$, $2006$, and $2008$ are all ultra-even years. They are all $2$ years apart, which is the shortest possible gap. $2009$ is not an ultra-even year because of the $9$, and $2010$ is not an ultra-even year because of the $1$.
a. In the years between the years $1$ and $10000$, what is the longest possible gap between two ultra-even years? Give an example of two ultra-even years that far apart with no ultra-even years between them. Justify your answer.
b. What is the second-shortest possible gap (that is, the shortest gap longer than $2$ years) between two ultra-even years? Again, give an example, and justify your answer.
a. In the years between the years $1$ and $10000$, what is the longest possible gap between two ultra-even years? Give an example of two ultra-even years that far apart with no ultra-even years between them. Justify your answer.
b. What is the second-shortest possible gap (that is, the shortest gap longer than $2$ years) between two ultra-even years? Again, give an example, and justify your answer.