unit fraction from consecutive amounts

combined events, without replacement

a ppt is here

with consecutive numbers of red and white counters (e.g. 8 red, 9 white)
when is the probability of obtaining two reds a unit fraction (1/k)?

it is reasonably easy to show that (n - 1) needs to be a factor of 6
this gives the four solutions, all of which are fairly small
so trial and improvement should obtain most if not all of the solutions