July 2024
Problem of the Month

Where do the Magicians Rank?
Submitted by K. Sengupta of Calcutta, India

Six magicians A, B, C, D, E, F live on 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.

Solution:

C-1, A-2, E-3, F-4, B-5, D-6

EXPLANATION:

From statement (2), F's ranking changed from 6 to 2 and D's ranking changed from 4 to 3 or F's ranking changed from 6 to 4 and D's ranking changed from 4 to 6.

If former were the actual case, then F's change would have been a four step change.
But, B's ranking could not have changed by more than four steps, so a four step change in F's ranking would lead to a contradiction of statement (1).
So, F's ranking changed from 6 to 4 and D's ranking changed from 4 to 6.

From statement (1), B's ranking changed from 2 to 5, a three step change.
Since E's change in ranking was smaller than B's, his ranking changed to 3.
By similar reasoning, A's ranking changed to 2 and by process of elimination, C's ranking was 1.

Correctly solved by:

1. Kamal Lohia	Holy Angel School, Hisar, Haryana, India
2. Colin (Yowie) Bowey	Beechworth, Victoria, Australia
3. Kelly Stubblefield	Mobile, Alabama, USA
4. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
5. Ivy Joseph	Pune, Maharashtra, India
6. Austin Hale	Grand Junction, Colorado, USA
7. Davit Banana	Istanbul, Turkey

Send any comments or questions to: David Pleacher