What is the minimum number of races required to pick the
best three horses from 25 horses if each race has maximum of
5 horses?
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Answer Posted / diepark
An explanation as to why the answer is 7, assuming no method
to time the races:
Divide the 25 horses into groups of 5 labeled A to E. Hold 5
races, one for each group and label the horses A1, A2, etc.,
where the number designates what place the horse finished.
Hold a 6th race out of the five winners of each group, and
arbitrarily pick A1 as the winner, B1 as second place, etc.
After the 6th race, you'll have the following tree for
fastest horse:
A1
| \
A2 B1
| | \
A3 B2 C1
| | | \
A4 B3 C2 D1
.
.
.
As you can see, A1 is the fastest horse and is faster than
all the horses in the other groups since he beat their
winners, and likewise, B1 is faster than the horses in
groups C to E since he beat their winners. But we don't know
if B1 is faster than the rest of the horses in group A,
since they never raced against each other, and likewise, we
don't know if C1 is faster than the remaining horses in
group B, since they didn't race each other, etc. To
determine the 2nd fastest horse, let's hold a hypothetical
7th race between A2 and B1. There are two possible outcomes
that result in the following trees:
A1 A1
| |
B1 A2
/ | \ / |
A2 B2 C1 A3 B1
| | | | | \
A3 B3 C2 A4 B2 C1
In the first outcome (left tree), B1 has won and is the
second fastest horse. Now we need to find out which horse is
3rd fastest in an 8th race between A2, B2, and C1, since
they have never raced each other.
In the second outcome (right tree), A2 has won and is the
second fastest. Therefore, an 8th race needs to happen
between A3 and B1 to determine the 3rd fastest horse.
If you consider which horses run in the hypothetical 7th and
8th races (horses A2, A3, B1, B2, and C1), you'll notice
that only 5 horses need to race, so all 5 horses can
actually run in the 7th race to determine the 2nd and 3rd
fastest horses.
So, is it possible to find the top 3 fastest horses in only
6 races? Yes, there are certain situations where this is
possible. Take the scenario where the winner always gets to
run in the next race: the first race has 5 horses (Group F)
that haven't raced; the winner goes on to race 4 more horses
that haven't raced (Group E), and that winner gets to race 4
more (Group D), etc. until the 6th and last race (Group A).
As long as the winner from the 5th race does not finish in
1st or 2nd, then you can determine the 3 fastest horses;
otherwise, you won't be able to determine the 2nd and/or 3rd
fastest horses, since the horses in group A have not raced
any of the other horses.
Race 1(F) 2(E) 3(D) 4(C) 5(B) 6(A)
----------------------------------------------------
1st F1-. E1----->E1-._ C1. B1-. A1
2nd F2 \ E2 D1 `->E1 \ B2 \ A2
3rd F3 `->F1 D2 C2 \ B3 `->B1
. F4 E3 D3 C3 `->C1 A3
. F5 E4 D4 C4 B4 A4
Thus, to guarantee you can determine the fastest 3 horses,
you need 7 races.
| Is This Answer Correct ? | 11 Yes | 2 No |
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