Question

In the table below is the list of teams, ranked from highest (#1) to lowest (#32), which are due to play in the Football World Cup. This tournament has four knockout rounds before the final, i.e. first round, second round, quarterfinals, and semifinals. In the first round, the highest ranked team plays the lowest ranked team (#32) which is designated as match no 1 of the first round; the second-ranked team plays the 31st ranked team which is designated as match no 2 of the first round, and so on. Thus, for instance, match number 16 is to be played between the 16th ranked team and 17th ranked team. In the second round, winner of match 1 of first round plays the winner of the match no 16 of the first round and is designated match 1 of the second round. Similarly, the winner of the match no 2 of first round plays the winner of the match no 15th of the first round and is designated as the 2nd match of the second round. Thus, for instance, match no 8 of the second round is to be played between the winner of match No.8 of the first round and the winner of match No.9 of the first round. The same pattern is followed for later rounds as well.
$\begin{array}{cccccc}RANK& TEAM& RANK& TEAM& RANK& TEAM\\ 1& Brazil& 12& Cameroon& 23& Croatia\\ 2& Italy& 13& Switzerland& 24& Guinea\\ 3& Argentina& 14& Uruguay& 25& Ghana\\ 4& France& 15& Ukraine& 26& Romania\\ 5& England& 16& USA& 27& Turkey\\ 6& Netherlands& 17& Denmark& 28& Ecuador\\ 7& Spain& 18& Mexico& 29& Egypt\\ 8& Portugal& 19& Paraguay& 30& Poland\\ 9& Germany& 20& Cote{d}^{\prime }Ivoire& 31& Tunisia\\ 10& CzechR& 21& Colombia& 32& Greece\\ 11& Nigeria& 22& Sweden& & \end{array}$
If, in the first round, all even numbered matches (and none of the odd-numbered ones) results in surprise win, and there are no surprise wins in the second round, then who would be the lowest ranked team facing Brazil in semi-finals?

• A. Switzerland
• B. Egypt
• C. Germany
• D. England

Answer 1

The correct option is A

Switzerland

Keeping both the rules in mind let’s try to get a rough idea about what’s the ideal scenario in the football world cup [the better ranked team beats its opponent, in other words- there are no upsets].

$\begin{array}{ccccc}\text{Rnd~1}& \text{Rnd~2}& \text{Rnd~3(Quarter)}& \text{Rnd~4(Semi)}& \text{Rnd~5(Final)}\\ \text{1) 1 v/s 32}& \text{1) 1 v/s 16}& \text{1) 1 v/s 8}& \text{1 v/s 4}& \\ \text{2) 2 v/s 31}& \text{2) 2 v/s 15}& \text{2)2 v/s 7}& & \text{1 v/s 2}\\ \text{3) 3 v/s 30}& \text{3) 3 v/s 14}& \text{3)3 v/s 6}& \text{2 v/s 3}& \\ \text{.}& \text{.}& \text{4) 4 v/s 5}& & \\ \text{.}& \text{.}& & & \\ \text{16) 16 v/s 17}& \text{8) 8 v/s 9}& & & \end{array}$
Did you notice something? The sum of ranks of each round is constant:

$\begin{array}{ccccc}\text{Rnd 1}& \text{Rnd 2}& \text{Rnd 3(Quarter)}& \text{Rnd 4(Semi)}& \text{Rnd 5(Final)}\\ 33& 17& 9& 5& 3\end{array}$
Let us call these numbers the magic numbers of the respective rounds.

Brazil (1) plays rank 4 in semi-finals (Round 4 – magic number is 5)

But, we know all even numbered matches of round 1 and all even ranked teams lose in round 1, hence let’s trace rank 4’s path to know which the LOWEST ranked team that can replace it is.
Round 1: 4 + 29 (“Magic number for round 1 is 33=4+29”) $\Rightarrow 29$ qualifies and takes 4’s place
Round 2: 4 + 13 (“Magic number for round 2 is 17=4+13”)
But, now the round 2 match is not 4 v/s 13 but 29 v/s 13 $\Rightarrow 13$ is a better rank team and no upsets occur in round 2 so 13 qualifies and takes 4’s place in the next round.
Round 3: 4 + 5 (“Magic number for round 3 is 9=4+15”)
But now here 4 is replaced by 13 so its 13 v/s 5. The winner of this match meets 1 in the next round (Semi-finals). We have the liberty to choose any of these two teams but the choice has to be of the LOWER team hence, choose rank 13 =Switzerland as the ANSWER