Hint: Combinatorics Q4

This is a classic case where looking at much smaller examples will be a big help. So, go away and write down all the ways of colouring:

The number 1 black or white

The numbers 1 and 2 black or white

The numbers 1, 2 and 3 black or white

And compute the associated numbers. When there's a unique odd value what is it?

Look at some random examples of slightly longer colourings (7, 8, 9 or 10). Consider some special cases (where the black and white pattern is simple). When we get a unique odd value is it always the same? In general, how do the values change as we move through the list from left to right? When do two numbers, particularly consecutive ones, get the same associated value? How can we manipulate colourings without changing the number of values that occur an odd number of times? Aim towards producing a simple colouring which has the same value repeated an odd number of times as a (possibly) complicated one does.

Go on, get your hands dirty -- you come to grips with problems of this type by learning the shape and feel of the problem -- not by attacking it head first.

Remember that when you have 100 keys one of which might fit a lock the main thing is to start trying them, and not to get discouraged when the first ten don't fit. You would never imagine that the solution to that problem was to go out and buy more keys!