## 50. Beauty Contest.

- Can you tell me in confidence, in what order did the five most beautiful girls take the first five places in the beauty contest organized by your magazine? - I asked the publisher. Of course, she refused to prematurely disclose the secret, but promised to tell me if I had guessed the places correctly.
- Did the places fall in the order $A-B-C-D-E ?-$ I asked.
- However, you are a master of missing the mark, - the publisher joked. - Not only did you not name the correct places for the participants, but you also did not guess the rival who took the immediately preceding place for any of them.
- Then, perhaps, the girls were placed in the order $D-A-$ $E-C-B ?-$ I asked.
- You are beginning to improve, - she cautiously encouraged me - You correctly indicated the place for two participants, and for two of them, you even correctly named the rival who took the preceding place.

After some calculations, I reported the correct order, and the publisher made me promise not to disclose it. How did the places among the five contestants of the beauty contest actually fall?
50. In the order $D-A-E-C-B$, the two participants who are in the correct positions must follow each other directly. Otherwise, since two participants correctly named their immediate predecessors (i.e., the girls who took the immediately preceding place), the correct positions would have been identified for three, not two, participants. This means that one of the pairs $D-A, A-E, E-C$, or $C-B$ must be correct. The pairs $A-E$ and $E-C$ can be discarded, as such combinations would not allow two participants to correctly name their immediate predecessors. If the correct pair were $D-A$, the order would have to coincide either with $D-A-B-E-C$ or $D-A-C-B-E$, leading to a contradiction with the comment made by the editor regarding the first guess. Therefore, the pair $C-B$ must be the correct one. The order $A-E-D-C-B$ should be discarded, again considering the comment on the first guess. Thus, the only possible distribution of places that satisfies all conditions is $E-D-A-C-B$.

[From J. F. Litch, A. M. M., 68, 669 (August 1961).]