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the BIG riddle topic [part IX]

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moogle
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PostPosted: Tue 09 Jan, 2018  Reply with quote

Alot wrote:
There are 25 race horses that each run at different speeds. You can only race 5 horses at a time


I'm tentatively thinking 7.
Divide horses into 5 groups.
1. Race the 5 sets of horses.
2. Race winners together.
3. Race the best 3 of winners race with runners up in fastest of the first 2 races.

kiekeboe I think I need to rethink this. Oh 8 races should be enough. whatsthat


Please pm me Alot before I go crazy..



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Yev
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PostPosted: Tue 09 Jan, 2018  Reply with quote

Alot wrote:
What is the minimum number of races you need to determine the fastest 3 horses?


I think 9 races.

The races are conducted in 3 rounds. The first round, the horses are divided up into 5 groups of 5 horses. Each group runs a race, making 5 races in this round.

The second round has 3 races. All of the first-place horses from each of the first-round races run against each other, all of the second-place horses run against each other, and all of the third-place horses run against each other. The fourth- and fifth-place horses are eliminated.

The horse that comes in first out of all the first-place horse is the fastest overall, and need not run in another race.

Out of the horses that came in first place in the first round, the ones that came in second and third in the next round go on to the final round. The fourth- and fith-place horses are again eliminated.

Out of the horses that came in second in the first round, the ones that came in first and second in the next round go on to the finals. The third place is eliminated because that horse had been beaten by 3 other horses already: one in the first round and 2 other horses in the second. Fourth- and fifth-place are eliminated, as before.

Finally, out of the horses that came third in the first round, only the horse that came first in the second round goes on to the finals. The other horses were already outrun by 2 horses in the first round, and 1 or 2 horses in the second round. Fourth- and fifth-place are eliminated, as before.

Thus, the horses that run in the final race are the second and third place out of the first-placers, the first and second out of the second-placers, and the first out of the third-placers. The winner and runner-up from this round are the second and third ranked overall, joining the winner of the first-placers' race in the top 3.

So, there are 5 races in the first round, 3 race in the second, and one final race, making 9 in total.


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Yev
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PostPosted: Tue 09 Jan, 2018  Reply with quote

Considering this further: No, I was wrong, and moogle's solution was closer than mine. It is 7 races.

As before, in the first round, each group of horses runs a race, making 5 races in the second round.

Then, in the next round, there is just one race, with the first-place winner from each group from the first round. As before, the first-place winner from this race is first-place overall, and need not run another race.

In the final round, both the second-place and third-place winners from the first-round group that the overall first place ran in need to run, because all of the top 3 might have ended up in that group.

The second place horse from the second round, and the runner up to that horse from the second round also need to compete in the final round. The lower places don't need to run, because the third-place horse is slower than the second-place horse from the first round, which is slower than the first-place horse from that round, which is slower than the first-place horse from the second round. Thus, we know there are 3 horses that are faster than those horses.

Likewise, the the third-place horse from the second round is, at best, third-place overall, so none of the runner-ups which that horse beat in the second round need to compete in the final round.

Therefore, in the final round, the 5 horses are the second- and third-place horses from the second round, the second- and third-place horses that the overall first-place horse outran, and the runner-up from the first-round to the horse that came in second-place in the second round. The top 2 horses from the final round are second- and third-place overall.

So, there are 5 races in the first round, one in the second round, and one in the final round, for 7 races in all.


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Alot
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PostPosted: Wed 10 Jan, 2018  Reply with quote

^this is the same answer I got, and I'm pretty sure it can't be done in 6.. (is there a mathematical way to prove this?)
@moogle, you were close, just need to rethink #3 a tiny bit wink5


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moogle
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PostPosted: Wed 10 Jan, 2018  Reply with quote

Alot wrote:
^this is the same answer I got, and I'm pretty sure it can't be done in 6.. (is there a mathematical way to prove this?)
@moogle, you were close, just need to rethink #3 a tiny bit wink5


Yes, dawned on me later. Fastest horse is found in race 6. So only 2nd and 3rd need to be found.

So race 7 needs
horse 2nd and 3rd from 'fastest horse' first race.
horse 1st and 2nd from 'second fastest horse' first race.
3rd horse in all winner race.



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Obliverum
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PostPosted: Sat 13 Jan, 2018  Reply with quote

horse race part 2 🐎🐎🐎

the manager of Timeless Horse Races Inc. thanks you all for helping with her horse racing problem. however she has one concern. after she knows which three horses are fastest she plans to have a big showcase race between them and she doesnt want anyone to know which horse will win beforehand.

can you give her a way to race the 25 horses to find the fastest 3 but without finding out which is the fastest of them all?

she also would like you to help her bookie set the odds for the showcase race.



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Alot
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horse race part 2 🐎🐎🐎
PostPosted: Tue 06 Feb, 2018  Reply with quote

From initial analysis, I would say it's not guaranteed to be possible to find the three fastest without knowing which is fastest, since you might get lucky (or unlucky) and race the three fastest in one of the initial races. However, the odds of that would be 231/53130 or approx 0.43% if I managed to get my probabilities right.. (22C2 over 25C5?)

Anyway, there's still a higher probability that it will be possible, so here's how I would propose to proceed:
Click here to see the hidden message (It might contain spoilers)


I made a spreadsheet to illustrate my solution, though it's not fully done yet, and only based on scenarios. Basically I numbered the horses from 1 to 25 with 1 being the fastest, and randomly grouped them for Round 1. Orange colored cells are the horse numbers. White colored cells refer to their respective rank from Round 1 races. Press F9 or edit any cell to generate another scenario.

Example scenario 1:
Click here to see the hidden message (It might contain spoilers)


Example scenario 2:
Click here to see the hidden message (It might contain spoilers)


Example scenario 3:
Click here to see the hidden message (It might contain spoilers)


--

EDIT v2 of spreadsheet here (with odds)
Click here to see the hidden message (It might contain spoilers)


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