# Three shy fairies performing two thousand times

Three shy fairies participate in a dance competition. Since they are shy they don't want to be seen by anyone except for the five judges of the competition. The competition is composed of $$2000$$ matches in which the fairies perform a short ballet in a tiny room and then each judge votes for the fairy that they liked most.

After each match the fairies exit the tiny room to share with the audience the results of the voting, but (since they are shy) they agreed to speak all together i.e. to communicate at the very same time the number of votes they achieved. Unfortunately in this way their voices mix up: for example if two fairies got one vote and the other fairy got three votes they say -with very low voices- "One", "One", "Three"; but the audience will just hear "One", "Three".

After having shared the results of the first match with the audience the fairies enters the tiny room again for the next match, then the five judges vote again, then the fairies exit again and communicate the results and so on for $$2000$$ times in total.

Adel, Barden, Carl, Deka and Ester are five members of the audience (although it isn't a proper audience since they couldn't see any of the ballets). After all the $$2000$$ matches

• Adel says: I have heard "One" $$380$$ times.
• Barden says: I have heard "Two" $$175$$ times.
• Carl says: I have heard "Three" $$194$$ times.
• Deka says: I have heard "Four" $$329$$ times.
• Ester says: I have heard "Five" $$1461$$ times.

With this information, can you tell:

1. how many matches ended in a draw?
2. how many times a fairy achieved zero votes in a match?

Please explain the procedure you used to solve this puzzle.

Source: I invented this puzzle out of voting data from a real experiment.

It seems rather straightforward to me, unless I've misunderstood some detail of the question.

There are only 5 possible ways that the 5 judges' votes can be partitioned into three numbers: $$a: 5+0+0 \\ b: 4+1+0 \\ c: 3+2+0 \\ d: 3+1+1 \\ e: 2+2+1$$

From what the audience heard, we have:

$$a=1461 \\ b=329 \\ c+d=194 \\ c+e=175 \\ d+e=51$$

This has the solution:

$$a=1461 \\ b=329 \\ c=159 \\ d=35 \\ e=16$$

The answers to the questions are therefore:

1. Only outcome $$e$$ is a draw, so there are $$16$$ draws.
2. The number of times a zero heard by the audience is $$a+b+c=1461+329+159= 1949$$, but case $$a$$ has two fairies that received zero so the number times a fairy got zero points is $$2a+b+c= 1949+1461= 3410$$.

• Correct, it was a simple puzzle. – melfnt Jan 27 at 10:42
• @melfnt I notice that you could have omitted one of the listener numbers, since we also have the fact that $a+b+c+d+e=2000$, which I did not use. – Jaap Scherphuis Jan 27 at 12:22
• You are right, I used another method to solve this puzzle while creating it. This method was far less formal and required all the information I gave – melfnt Jan 27 at 13:21