In problem 194, we see the solution for the 2-number case is 1/3. The following shows the solution for the 3-number case is 1/4, and for the 4-number case is 1/5. Following the pattern, the solution for the n-number case is 1/(n+1).
If it isn't clear from the scan below, first I find the area of integration, and then multiply the expected average by the inverse of that area. In the 3-number case, x is the largest number, and ranges from 0 to 1. y is the second largest, and ranges from 0 to x. z is the smallest and ranges from 0 to y. Same concept in the 4-number case, but we start with w as the largest number.
Michael Shackleford, A.S.A.