## Sunday, 20 November 2011

### Puzzle-24 (CAT-2008)

Directions for Questions 1 and 2:
Choose (1) if Q can be answered from A alone but not from B alone.
Choose (2) if Q can be answered from B alone but not from A alone.
Choose (3) if Q can be answered from A alone as well as from B alone.
Choose (4) if Q can be answered from A and B together but not from any of them alone.
Choose (5) if Q cannot be answered even from A and B together.

In a single elimination tournament, any player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules:
(a) If the number of players, say n, in any round is even, then the players are grouped in to n/2 pairs.    The players in each pair play a match against each other and the winner moves on to the next round.
(b) If the number of players, say n, in any round is odd, then one of them is given a bye, that is, he automatically moves on to the next round. The remaining (n − 1) players are grouped into (n − 1)/2 pairs. The players in each pair play a match against each other and the winners move on to the next round. No player gets more than one bye in the entire tournament.

Thus, if n is even, then n/2 players move on to the next round while if n is odd, then (n + 1)/2 players move on to the next round. The process is continued till the final round, which obviously is played between two players. The winner in the final round is the champion of the tournament.
1.  Q: What is the number of matches played by the champion?
A: The entry list for the tournament consists of 83 players.
B:  The champion received one bye.
2. Q: If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?
A: Exactly one player received a bye in the entire tournament.
B:  One player received a bye while moving on to the fourth round from third round

Solution:

Q1)
To find the number of matches played by the champion:
Statement-A specifies that the entries are 83.
64 < 83 < 128 => 26 < 83 < 27 => the number of rounds = 7
 no. of players Round-I Round-II Round-III Round-IV Round-V Round-VI Round-VII 83 41+1 21 10+1 5+1 3 1+1 1
(Observe that ‘bye’s are indicated in colour)
If the champion has not received any bye, then he has to play 7 matches,
But if he has received a bye, then he has to play 6 matches. Hence it is not possible to decide it from the statement A alone and statement B is also required to answer the question.

Q2)
The numbers when repeatedly divided by 2 – always leaving remainder zero until it ultimately becomes 1 are the powers of 2:
20,21,22,23,24,25....
ie., 1,2,4,8,16,32,64,.....
The numbers when repeatedly divided by 2 – always leaving remainder zero but exactly one time leaving 1 until it ultimately becomes 1 are in the form of (2n-1)2k, where n > 1, k ≥ 0 and n,k being positive integers:
3,7,15,31,63,.....
and  3*2 = 6, 3*4 = 12, 3*8 = 24,.....
and  7*2 = 14, 7*4 = 28, 7*8 = 56,.....
and so on...
Now this set of numbers is to be considered here for this problem as it is mentioned that “Exactly one player received a bye in the entire tournament
As “the number should be between 65 and 128”, the possibilities are:
3*32 = 96;     7*16 = 112;   15*8 = 120;   31*4 = 124;   63*2 = 126;   127*1 = 127
Now we check up case by case for the occurrence of ‘bye’s:
 no. of players Round-I Round-II Round-III Round-IV Round-V Round-VI Round-VII 96 48 24 12 6 3 1+1 1 one player received bye while moving on to 7th round from 6th round 112 56 28 14 7 3+1 2 1 one player received bye while moving on to 6th round from 5th round 120 60 30 15 7+1 4 2 1 one player received bye while moving on to 5th round from 4th round 124 62 31 15+1 8 4 2 1 one player received bye while moving on to 4th round from 3rd round 126 63 31+1 16 8 4 2 1 one player received bye while moving on to 3rd round from 2nd round 127 63+1 32 16 8 4 2 1 one player received bye while moving on to 2nd round from 1st round
(Observe that ‘bye’s are indicated in colour)
Now it is evident that, occurrence of one-time bye can happen in six cases. Hence it is not possible to decide it from the statement-A alone and statement-B is also required to answer the question.