Solution to 9):

Impose a numbering of the boxes themselves. The numbers inside the boxes are a 1-1 map onto the boxes. If you draw this 1-1 map as a directed graph you will get a distribution of cycles. In order to find the box that maps to your number, start at the box of your number and open the box it refers to, then open the box which that box refers to, etc. You will only fail if the cycle is larger than 50. Hence if no cycle is larger than 50, all logicians will find their own number. The probability of having a cycle of size j greater than 50 is (j!/j)(100 choose j)(100 - j)! = 1/j. Hence the total probability is (1/51)+(1/52)+...+(1/100) = 0.6882, so that the toal probability of survival is 31.18%.