Six magicians A, B, C, D, E, F live in an island. Every Monday, the elders of the island rank the six magicians on their performance during the previous six days.

The rankings published on the previous Monday listed A 1st, B 2nd, C 3rd, D 4th, E 5th and F 6th.

This Monday’s ranking has each magician ranked in a different position than that of the previous Monday. We know that:

1. B’s change in ranking is greatest among the six magicians.

2. The product of D’s rankings for the two weeks is the same as the product of F’s rankings for the two weeks.

Determine the new rankings.

D is in 4th position (2*2) and F is in 6th (2*3). The common product of their positions must therefore be a multiple of 2*2*3 = 12. D could have moved from 4 to 3 while F moved from 6 to 2, but then F would have moved at least as far as B could, and B is presumably not merely tied for first place in moving positions.  The other possiblity, then, is that D and F switched places, with D moving from 4 to 6 and F from 6 to 4.

In order for B to be the farthest moving, he must now be in 5th place.

That leaves A, C and E to occupy the first three positions.  As B moved 3 positions, the most E could move is 2, putting him in 3rd place.  Since A can't still be in 1st place he must be in 2nd while C moves to 1st.

So the new sequence is C, A, E, F, B, D.

Posted by Charlie on 2007-05-08 11:48:05