**What is the exponent of the largest power of two whose base seven representation doesn't contain three zeros in a row?**

Not as easy as it sounds, Python probably unqiuely suited to this problem, this is because most programming languages have a mimited set of integer types, for instance C/C++ typically has

8 bit integers

16 bit integers

32 bit integers

64 bit integers

If you need anything else this needs to be programmed around. This is true for most languages, they have a set of limited integers, however Python has 2 types of integer

32 bit

any size you want

The second type clearly gives a very large problem domain, if you look at the base 7 representations of the first 31 powers of 2 (numbers fitting into a signed 32 bit int) none of them have 3 consecutive zeros.

The first question has to be

Are there any powers of 2 which have 3 consecutive 0s in their base 7 representation?

Assuming the answer to that is yes the question then makes an assumption:

At some value N the base severn representation of 2 ** x for all x >= N contains at least 3 consecutive 0s.

This statement is not self evidently true to me however assuming that it is true then you need to find the minimum value of N and then the answer will be N-1.

Personally I would say thorum solving like this involves mainly maths with little need for programming. In fact I think it would be very hard to write a program to some this to be true, you need to some something to be true for all x >= N not something a computer is very good at doing. Computers are good at doing something for all MIN_LIMIT <= x <= MAX_LIMIT but for this problem there is no max limit, a mathematical proof is required.