Without loss of generality, we can assume that the left piece L is smaller or equal to the right piece R. Hence, we are looking for the average X = R/L. Clearly, if we break a match in the middle, the ratio is 1/1. However, as our break point moves further to the left, the ratio changes (but it is equally likely that we will break the match there). For example, R/L is 3/1, 7/1, and 15/1 if we break the match at 3/4, 7/8, and 15/16 respectively. In other words, the ratios get large very quickly as we get closer to the edge, i.e. where L<<R. We can create two series that bound our desired value X from above and below. For the lower sum, we can keep going with the sequence mentioned above to infinity. However, in each interval we assume that the ratio is the same as the previous discrete point we encountered. In other words, between the middle of the match and 3/4 of the match, we approximate the ratio as 1/1. From 3/4 to 7/8, we approximate the ratio as 3/1, etc. The result is obviously smaller than our desired average ratio X. Similarly, we can form an upper sum by assuming that the ratio between the discrete points of the series is the next higher ratio. For example, we approximate the ratio between the middle of the match and 3/4 of the match to be 3/1. From 3/4 to 7/8 it's 7/1, and so on. These two series bound the average ratio from above and below:We can simplify the left hand side (lower bound) as follows:The second sum converges to 1 but the the first sum diverges. However, since our lower bound of X diverges, X iself, i.e the average ratio of the length of the longer piece to the length of a shorter piece diverges as well (is infinity). In particular, this means that a monte carlo simulation would yield wildly different results with every run.