horses.md

5th round (5x5x5x5x5)

If we split the 25 hourses into groups of 5 and have each group race among themselves, we'd be able to establish the ranking inside each group:

group:        1 2 3 4 5
1st finisher: a b c d e
2nd finisher: f g h i j
3rd finisher: k l m n o
4th finisher: p q r s t
5th finisher: u v w x y

With this we can elimite the 4th and 5th placers from each group because it is obvious that there are atleast 3 horses faster than them.

Eliminated: p q r s t u v w x y (10 horses, 15 left)

Note that we cannot eliminate across groups because we do not have ordering information about horses between groups just yet. So the 5th placer in one group might actually be faster than the 1st placer in another group.

6th round

Now that we have intra-group ordering information, the next step is trying to establish inter-group ordering. So we have the 1st placers from each group race each other. We use the 1st placer because they provide the highest number of eliminations (since there are atleast 2 horses that are slower than them.)

Race: a b c d e

a1 b2 c3 d4 e5
f  g  h  i  j
k  l  m  n  o

Assuming finishing placements as tagged above, we can eliminate 9 more horses which have evidence that there are atleast 3 horses faster than them:

Eliminated:

• d, e, i, j, n and o - because the fastest in their group is just eliminated.
• h and m - because they are slower than c and thus slower than b and a as well.
• l - because he is slower than g and thus slower than b and a

7th round

With the previous round we are left with 6 horses:

a b c
f g
k

Summarizing what we know so far:

• a > f > k
• b > g
• a > b > c
• thus a > [b, c, f, g, k]

Now from the above we can infer that a is definitely the fastest horse. And we are left with exactly 5 horse with partial ordering information so we just need one more race to determine the 2nd and 3rd placers.

So assuming the resulting placement is b > c > f > g > k, we now have all information necessary to determine the 3 fastest horse:

• a > b > c > f > g > k

And we are done.