A Reader's Solution
Here is Rafik Ouared's solution to puzzle number 6.
The answer to the question is : 81 medallions on average (rounded number).
Explanation: the equivalent problem is to fill 22 empty boxes sequentially. Each time one fills one box, he moves to fill the next one by drawing in average a number of medallions before finding the one different from the ones in the boxes behind.
Let N be the number of boxes to fill (N=22).
Let n be the position of the nth box to fill after the n-1 before filled.
Let M_n be the average of medallions one needs to fill the nth box with the right medallion.
Let P_n the probability to find the right medaillon for the nth box.
P_n = 1/( M_n ) = (N-n+1)/N
So, for the nth box, the average number of medallions one needs is:
M_n = N/(N-n+1)
And to get N=22 different pictures one needs in average to buy :
M = M_1 + M_2 +....+ M_22 = 22*(1/22 + 1/21 + 1/20+...+1/2 + 1)
M = 81.1978 rounded to 81.