A tournament in the planet Lynn pitted each elf contestant against every human and Minotaur contestant, each human contestant against every elf and Minotaur contestant, and each Minotaur contestant against every elf and human contestant in three contests of skill: wrestling, archery and knife throwing. In other words, each type of being- elf, human and Minotaur- competed against everyone in the other groups of beings in three different contests.

These facts are known:

1. The total number of human and Minotaur contestants is nine.
2. The number of elf vs. human competitions in the three skills was seventy-two.
3. Twenty-seven contests were held each day.

How many days did the competition last, given that it lasted for less than 30 days?

For the following M is the number of Minotaurs, H is the number of humans, and E is the number of elves.

We can extropolate from (2.) E × H × 3 = 72, and E × H = 24.
The divisors for 24 are 1, 2, 3, 4, 6, 8, 12, and 24 giving the
human and elf pairs (H, E) as:
(1, 24), (2, 12), (3, 8), (4, 6), (6, 4) (8, 3), (12, 2) and (24, 1).

We are given from (1.) that M + H = 9. Thus, the number of humans can not be greater than 9, and we can reduce the possible (H, E) pairs to:
(1, 24), (2, 12), (3, 8), (4, 6), (6, 4) and (8, 3).
And we can calculate the Minotaur-Human pairing to be
(1, 8), (3, 6), (4, 5), (6, 3), (7, 8) and (8, 1).

Where C is the number of contests,
C = 3×((M × H) + (M × E) + (H × E)).

M   H   E   C
1   8   3  105
3   6   4  162
4   5   6  222
6   3   8  270
7   2  12  366
8   1  24  672

From (3.) we are given that C/27 should be an integer. Only 162 and 270 are divisible by 27, both of which provide a length of days under 30.
Where the number of Minotaurs, humans and elves are 3, 6 and 4 respectively, the 162 contests lasted 6 days.
Where the number of Minotaurs, humans and elves are 6, 3 and 8 respectively, the 270 contests lasted 10 days.

Edited on July 26, 2010, 7:02 pm

Posted by Dej Mar on 2010-07-26 18:54:36