I was given this problem, printed on a T-shirt handed out at a math conference.
Below is my answer to the first part, which I believe is correct.
Does anyone know the answer to the second question as well?
The defined structure is a ‘binary’ fractal because it divides into two parts each time.
The first question is to find the maximum expected value for the average of the values encountered when traversing the entire structure, as the number of subdivisions approaches infinity, if the values 0 or 1 are randomly assigned to each small square. That is, the maximum average value obtainable by following one of the paths.
[One path could be, for example, 1+0+1+0+1+0+0+1+1+1+…, another 1+1+1+1+1+1+1+1+1…]
Since the subdivision is binary, the chance of getting a 1 each time is 1/2 (one in two). But the number of paths grows at the same rate at which the probability decreases (doubling at each step), so it should always be possible to find a path where all the squares contain 1s; therefore, I would say the answer for the maximum of this expected value is “1”.
The following “what if” question, however, asks for the same expected value not for two discrete values, {0, 1}, but for the continuous values of the interval [0, 1] …
Not trivial… the chances of obtaining a number are no longer as simple as the 1 in 2 from before…
