As part of the application to the Wolfram High School Summer Camp (see this post for more information)
we have a problem set to allow applicants to distinguish themselves. I can't post them all here because if anyone answered on the Community, it would be too easy to copy, but here are a couple which it is safe for people here to discuss.
What do people think of these questions?
In base 3, the digits are 0, 1, and 2. For example, 2^32 in base 3 is 102002022201221111211. What is the least common digit in base 3 for the powers of 2? Make an educated guess and explain your reasoning.
Someone gives a coordinate somewhere on the Earth to Wolfram|Alpha. Wolfram|Alpha then automatically decides on a zoom level for this location on a map. For instance, if the location is in the ocean, the resulting map wouldn't just be all blue. Describe a heuristic to pick a zoom level automatically for any given location.
I guess for any base other than 2^n (n integer), the numbers will be 'smoothly' spread out over the digits. For bases 2^n, doubling (powers of 2) is nothing but doubling the first digit, which results in lots of zeros at the end.
Depending on the location:
if it is a city, you want to see the entire city with some padding.
If it is a road, you want to see the entire road.
If it is a crossing, you want to see both roads,
Same holds for states, counties, countries, continents.
Giving a GPS coordinate is much more subtle though. You probably want to find the nearest n-cities/rivers, and compute the maximum distance for that. If that exceeds the size of the country, just zoom to the country.
If the point is on water (oceans), you probably want to zoom as to see the containing ocean entirely...