Eight options with six sides
This one is based on a problem of the week somewhere.
The problem with great restaurants is that they often have so many delicious menu items that making a decision can be near impossible. For this reason, Kirk always brings his trusty 6-sided die with him when he goes out to dinner. But one fateful night, Kirk found no fewer than eight dishes he wanted to try. How could he use his die to fairly decide between his eight options? What is the least number of die rolls he would have to make in order to guarantee he can fairly pick on of the eight dishes?
What if he had nine options? What about other numbers of options? That is, for which number of options is there a way for Kirk to fairly decide between those options using only his 6-sided die. If $n$ is a possible number of options, what is the least number of rolls needed to decide?