10
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Six musicians are participating in a music festival. During each concert, some of the musicians perform, while the others listen in the audience.

What is the fewest number of concerts needed so every musician has listened (as an audience member) to every other musician perform?

Source: Aust MS Gazette, Volume 42, No. 2, in the Puzzle Corner section. See here.

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  • 2
    $\begingroup$ I suppose that playing in a very (VERY) large room, then sitting down and listening to the echo of the concert would be cheating? $\endgroup$ – Floris Jul 11 '15 at 3:59
15
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I believe I have it

Four concerts!

 Concert format: --- are playing for xxx
 --- xxx
 Concert |Done performing |Done listening
 abc def |                |
 cde abf |C               |          F
 aef bdc |AE              |         DB
 bdf cae |BDF             |        CAE
 
 My newest reasoning: 
 This problem is about relationships 
 In a 1 performing for 5, 5 relationships are formed, X is heard by abcdef 
 In a 3 performing for 3, 9 relationships are formed 
 a is heard by def | b is heard by def | c is heard by def
 so 3x3 is more efficient achieving the number of relatioships needed 

 So there are 6 people that need to be heard by 5 people, this is 30 relationships
 To do this in 3 turns we would need to generate 10 relationships per concert.
 Looking at the concert options there are only 0x6, 1x5, 2x4, 3x3 and symmetric concerts
 which are the same production of relationships.
 These produce, 0, 5, 8, and 9 respectively. 
 None of these are greater than 10 (Last I checked)
 However, very interestingly this means that it is potentially possible that
 2x4 could produce a solution, however this does not prove it possible as there could
 be duplicate relationships forced.
 But it does prove 4 concerts in the minimum.

Thanks to mmking for format help

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  • $\begingroup$ Working on formatting $\endgroup$ – Going hamateur Jul 10 '15 at 18:09
  • $\begingroup$ Who's playing, and who's listening? $\endgroup$ – itriedacrab Jul 10 '15 at 18:11
  • $\begingroup$ Fun fact, it doesnt matter, the activities are Identical, but I intended left performing, right listening $\endgroup$ – Going hamateur Jul 10 '15 at 18:11
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    $\begingroup$ This HAS to be optimal. Unless we count recording all 6 of them in one performance, and then letting all 6 watch that video, reducing it to 1 performance! $\endgroup$ – JLee Jul 10 '15 at 18:16
  • $\begingroup$ Very nice! If you prove your solution is optimal, I will accept your answer. $\endgroup$ – Mike Earnest Jul 10 '15 at 18:21
6
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Unless I'm missing something, the answer is

4


abc def
ade bcf
fcd abe
fbe acd

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  • $\begingroup$ Alternatively, one listens and the rest play $\endgroup$ – Spencerkatty Jul 10 '15 at 17:54
6
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Performing Audience

P    A
abcd ef
cdef ab
abef cd

fbd eac
eac fbd

5 performances but I think it can done in less.

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  • $\begingroup$ Nice! Is this optimal? $\endgroup$ – JLee Jul 10 '15 at 17:59
  • $\begingroup$ Not sure haven't looked for c=4 $\endgroup$ – Bob Jul 10 '15 at 18:00
  • $\begingroup$ Yeah, If I was a faster poster I would have had something similar to this I was think 2 v 4 and then you just need to get each to play for their partner, I had not yet stumbled upon the 3v3 idea. Well done $\endgroup$ – Going hamateur Jul 10 '15 at 18:00
6
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Solutions for four have been posted.

Three is impossible, because each musician must either:

  • play alone (and have all 5 listen)
  • play at least twice (if they are not playing alone, they must play for musician(s) playing with them the first time).

They must also either

  • be in the audience alone (and listen to all five)
  • be in the audience at least twice (they must listen to the person who was in the audience with them the first time)

For everyone to listen twice and play twice, 24 'slots' are needed, but only 18 are available.

If one plays alone, then in the remaining two concerts, the 5 others must all play twice, which means they cannot listen again.

If one listens alone, then in the remaining two concerts, the 5 others must all listen twice, which means they cannot play again.

Therefore it is impossible with three concerts.

There is no solution for 4 concerts other than for them all to be 3 vs 3, because:

Nobody can listen to anybody more than twice or play for anybody more than twice, there are only 24 slots, and we've already established 24 are required.

So suppose ABCD played to EF.

A would still need to play to B and C and D, and EF cannot listen to A again, so AEF to BCD is the next concert.

B also needs to play to ACD. Now E and F have both played to C and heard C, so cannot do either again without going over 24 spaces.

So no 4 vs 2 concerts.

It does not matter in fact whether E was listening in the first concert or playing, so the 5 vs 1 concert is out as well.

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2
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OK, I will go ahead and kick things off with

6, as follows.


enter image description here

I'm still trying other configurations to see if this is optimal.

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