here is the problem


The following problems require you to make some assumptions. Be sure to specify them.

What is the shortest the address on a typical letter could be, if it is to get to a unique human recipient? (Assume the permitted characters are A-Z, 0-9). Explain your assumptions and show your work. You will need a calculator that can do logarithms.

Please, help - I don't even understand what logs would have to do with this problem.

Thanks, Vertica

ya, me neither. If there are 4 billion people you need to support 4 billion addresses or one 32-bit integer (convienient!)

So, if the chars are A-Z and 0-9 you have a base-36 numbering system to work with.

How many digits of base-36 numbers do you need?

(in base-36, 10 == decimal 36, 100 == decimal 36*36, 1000 == 36*36*36)

I suppose you could use logs to aid in figuring that out, but it seems pretty simple.

That's exactly the question that logarithms will answer. Were invented to answer.

Going back to the definition: a^x=b --> log_a(b)=x, we have 36 as the base, and we know the result of the exponentiation is something in the vecinity of 4B. (4 Gigapeople??)

So a=36, and b=4,000,000,000. The equation on the left can't be solved directly for x, but the log identity gives an equation that can.

ceil(log_36(4,000,000,000)) = the minimum number of symbols chosen from a 36-member set, needed to uniquely identify one element out of a list of 4 billion items.

Sure, that result can be obtained without too much hassle by guessing, but it's faster AND THE WAY THEY WANT you to do the problem : )

A few assumptions I can think of:
-we know how many people there are on the planet
-there are 4 billion
*the number will not grow beyond 36^(ceil(log_36(4B))) during the expected system lifetime of our new addressing system
*addresses will be assigned to all humans without collisions
-a typical human recipient will be able to correctly identify their own unique address under the new system
-a typical human recipient will be able to REMEMBER their own unique address
*some mail routing/delivery system will be implemented that is compatible with the new unique-human-ID addressing scheme.
and finally,
-enough people will be able to be convinced that replacing the old system (name, city address, state, country) with the new system (a 7-digit address) is a good idea.