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Three Skill Settlement (Posted on 2010-07-26) Difficulty: 2 of 5
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?

No Solution Yet Submitted by K Sengupta    
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An unlikely, but possible interpretation | Comment 6 of 7 |

One way the problem can be interpreted is that each of the contests were not each contestant of one type against each contestant of another type, but each contestant of one type against every contestant of the other types. (A one to many relationship).

We then have the coefficients for the following groups:

E - the number of individual elves
H - the number of individual humans
M - the number of individual Minotaurs
1 - (E&H) the group of every elf and human
1 - (E&M) - the group of every elf and Minotaur
1 - (H&M) - the group of every human and Minotaur

If we interpret that the elf vs. human competitions are (E vs. H&M) and (H vs. E&M), such that there is at least 1 human in the group H&M, then from (2.) we have:
(E * H&M * 3) + (H * E&M * 3) = 72
(E * 1 * 3) + (H * 1 * 3) = 72
(E + H) = 24

From (1.), we have:
(H + M) = 9

We then have the table of possibilities:
                           Contest
 E  H  M  EvM&H HvE&M MVE&H Total
15  9  0    45    27    0    72
16  8  1    48    24    3    75
17  7  2    51    21    6    78
18  6  3    54    18    9    81 <-- Only total
19  5  4    57    15   12    84     divisible by 27
20  4  5    60    12   15    87
21  3  6    63     9   18    90
22  2  7    66     6   21    93
23  1  8    69     3   24    96

From this interpretation the competitions lasted only (81/27) = 3 days.


  Posted by Dej Mar on 2010-07-27 12:11:15
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